How does one go about understanding the poem "Remember"?

It is tempting to interpret, as opposed to analyze, poetry text from the perspective of one's current historical, social, or cultural era and personal experience (interpretation can often reduce to extrapolations of meaning from one's own personal experience). While this might be useful with some texts, perhaps with Contagion by Robin Cook or Extremely Loud & Incredibly Close by Thomas Schell, it is counterproductive with other texts, especially so with works having universal themes.
It can also be counterproductive to finding understanding to analyze texts without due attention carefully paid to syntax and punctuation structures: a semicolon versus a colon can radically alter the meaning of written text; this is why writers use them with such selective care (although styles of punctuation alter periodically and must be understood within their historic time frame to avoid confusion).
Christina Rossetti has been generally acknowledged by readers and critics alike to write "poetry displaying a perfection of diction, tone, and form" (PoetryFoundation.org). It is also acknowledged that Christina wrote on recurring ideas of the "inconstancy of human love, the vanity of earthly pleasures, renunciation, individual unworthiness, and the perfection of divine love [as] recurring themes in her poetry" (PoetryFoundation.org).
Thus, in consideration of these two aspects of her work--structure and theme--in order to fully understand Rossetti's work a two-fold approach to analysis is needed. Bear in mind that "analysis" can be very different from and yield very different results to "interpretation." Interpretation of a literary work may arise from one's personal experience, one's culture, one's society, one's historical period, or one's emotional response (which may or may not be related to direct personal experience, for example the emotional response elicited when watching The Blind Side though most of us are not underprivileged high school football players). Examples of these kinds of interpretation might be reading Oliver Twist through your own experience of poverty; reading "Remember" by Christina Rossetti through your own cultural norms; reading The Red and the Black through your own social-religious structure; reading Pride and Prejudice through the eyes of a 19th century Marxist; or reading Evelina by Fanny Burney according to the emotional response elicited, which may or may not relate to personal experience.
In general, analysis of literature applies more structured application of logic than these interpretive approaches do, although there are literary criticism approaches that incorporate interpretive elements, such as Reader-Response Criticism and Marxist Criticism. What is the advantage of either analysis or interpretation over the other?
The advantage of analysis over interpretation is that analysis helps us understand, as much as possible, what the author was expressing. Some think this is the most advantageous way to know what literature illustrates (illustrates rather than suggests) about human nature, the pitfalls of living, the nature of life, the pain of death, and metaphysical hopes.
To illustrate this concept of advantage, if Crime and Punishment is about a student who becomes obsessed with the philosophical notion of the "superman" who is above the regulating norms of life, that has little bearing on the majority of people that ever were or ever will be since most of us are not thus obsessed. Yet if Crime and Punishment is about a student who has an altruistic heart motivated by pity and anguish who literally and figuratively throws himself into his neighbors' fires thereby depriving himself of what is required for the sustenance of life and sanity, that could very well be applicable to many overly-large-hearted students anywhere in the world. Analysis rather than interpretation will identify which of these themes is the driving force in the text.
In order to accurately understand the complex form and structure of "Remember," a two-pronged analysis approach is needed to identify theme and meaning. For meaning, the close reading and formal analysis of New Criticism is needed to understand diction, syntax and the distinctions in meaning made by punctuation. For theme, the application of the principles of Historical Criticism (also called Traditional Criticism), which keeps touch with the author by examining biography (also society and culture if applicable), is needed.
Christina Rossetti Biography and Theme Notes
(Historical Criticism)
For the purposes of understanding "Remember," the salient parts of Christina Rossetti's biography are health--hers, her brother's and her father's--and the recurrence of themes in her corpus of work. Here, the job of the literary analyst is to understand the implications of life events and the significance of the dominance of some themes and the absence of others.
HEALTH
Health seems to have been an issue with some in the Rossetti family. Gabriel, Christina's father--a librettist, poet and Latin scholar--suffered a collapse of health in 1843, when Christina was thirteen. Her brother, Dante Gabriel, the poet and Pre-Raphaelite painter, later became dangerously ill in 1872, an illness that had devastating psychological components. Christina herself became seriously ill for the first time in 1845 after two years of helping to care for her father; she was fifteen. Her brother, William, an art and literary critic and editor, writes in his memoirs that understanding Christina's work requires understanding how seriously poor health impact and plagued her life, not once but repeatedly:

"Rossetti had bouts of serious illness throughout her life; William insists in his memoir that one cannot understand his sister unless one recognizes that she 'was an almost constant and often a sadly-smitten invalid.' The morbidity that readers have so often noted in her poetry, William suggests, was attributable to Christina’s ill health and 'the ever-present prospect of early death....'" ("Christina Rossetti," PoetryFoundation.org)

As literary analysts, we are well advised to understand the nature of Christina's life, full of the suffering of ill health as it was. We are well advised to take into account the instances when her health forced--or even seemed as though it would force--her body and mind to turn toward the portal of death only to be allowed by recovery to turn again toward living. Her first collapse in 1845 led to heart symptoms and depression (biographers understand little about the this first event).
Between 1870 and 1872, Christina suffered the worst of the bouts of illness she experienced (ironically, 1872 being the same year Dante suffered the turning point of his decline). She was at last diagnosed with Graves disease, a condition of the thyroid gland, after much suffering and many approaches toward death. Since "Remember" was published in 1862, ten years earlier, this particular incidence has no bearing on understanding this poem, yet it does illustrate and corroborate William's assertion that understanding Christina requires understanding the effects of her ill health on her everyday life beginning from her fifteenth year in 1845.

[1870-1872] "Her hair fell out, her skin became discolored, her eyes began to protrude, and her voice changed. After some months her doctors diagnosed a rare thyroid condition, exophthalmic bronchocele, more commonly known as Graves’ disease. Although Rossetti recovered, the threat of a relapse always remained. Moreover [according to William], the 'crisis left her appearance permanently altered and her heart weakened.'” (PoetryFoundation.org)

THEMES
Rossetti's first collection of poetry Verses appeared in 1847, two years after her first major health collapse, and had distinctly recognizable themes, particularly an emphasis on mortality and corruptibility. This is understandable since for four years she had been personally, physically and psychologically preoccupied with illness, the potential for death, and loss (loss because of the financial ruin the family fell into with Gabriel's illness and inability to work, although William and sister Maria helped mother Frances try to fill the financial gap).
"Remember" appears in Rossetti's second collection Goblin Market and Other Poems, which came out in 1862, the same approximate time that she began a tentative romance with Charles Bagot Cayley, one of her father's former students. Two things are of note: (1) this second collection continued the themes of death and corruptibility and (2) part of the reason her romance with Charles was tentative is that for more than twenty years she had been plagued with horrible bouts of illness and depression attending it.
Christina might well have had occasion to think that a romance with Charles was doomed to end in the grief of parting through death. Charles did propose in 1866, and Rossetti refused him. William states that, being very reticent, she never discussed the matter but that she did give the reason as being on "'on grounds of religious faith.'" (PoetryFoundation.org)
This significant event, though occurring after the writing of "Remember" published in 1862 (the approximate year of the commencement of their romance), relates to understanding Christina's psychological framework, which relates to understanding the overall meaning of "Remember." Of all Rossetti's poetry, from the 1847 collection Verses to all that she published up until the end of her life, more than half is religiously devotional in nature, indeed, all in her later years is religiously devotional.
Indicators of Context
In this context--a tentative romance with marriage refused on religious grounds and a poetess of devotional works--how likely is it that Rossetti would write a poem about ambiguous love and the rejection of a lover who will probably not regret her loss very much?
In this biographical context and in the context of a corpus of work that is serious, not frivolous, and that has corruptibility and mortality as the thematic "keystone" (PoetryFoundation.org), it is highly unlikely. In fact, it seems highly doubtful that she would have had the inclination or experience to write a poem about an insincere, ambiguous love that is broken off in life to avoid breaking it off in death. Such a poem would be inconsistent with her biography, with the themes that recur in her corpus and with the devotional nature of her psychology and poetic expression.