How is anachronism used in "Araby" by James Joyce?

Anachronism occurs when events in a story are portrayed in a way that is not chronologically accurate (when the sequence of events is told out of order, for instance). Flashbacks are a perfect example of this; they break up the linear chronology of the story. In "Araby" by James Joyce, anachronism is used throughout the first half of the story before giving way to a specific, linear narrative.
The first paragraph starts everything off by describing the setting in an ambiguous manner:

North Richmond Street, being blind, was a quiet street except at the hour when the Christian Brothers' School set the boys free. An uninhabited house of two storeys stood at the blind end, detached from its neighbours in a square ground. The other houses of the street, conscious of decent lives within them, gazed at one another with brown imperturbable faces.

There is no specific time frame or reference to a season. The only thing the reader knows is that the story appears to be set in the past, almost as if it were written as a memory. The next paragraph goes back and forth between describing a priest who died prior to the narrator moving into the house and some of the narrator's more recent exploits. It is only once we get to the third paragraph that we are given any reference to when the story might be set:

When the short days of winter came dusk fell before we had well eaten our dinners. When we met in the street the houses had grown sombre. The space of sky above us was the color of ever-changing violet and towards it the lamps of the street lifted their feeble lanterns.

The reader gets the sense that the story will be set at some point during the winter months. The narrator's activities are again described in a general sense. There is no specific day or time that the story is taking place; it simply gives a history of what the narrator would typically do during this time period.
The story then gets slightly more specific: "Every morning I lay on the floor in the front parlour watching her door." Again, there is no specific time setting, just the idea of a daily routine. Joyce does this as a way of giving a wide variety of background information in order to set up the latter part of his story. The author then continues the "routine" narrative structure to showcase the length of time in which the narrator has been infatuated with Mangan's sister. "Every morning" turns into "Saturday evenings," giving the impression that this has lasted for many weeks. It is not until the sixth paragraph that a specific time is outlined:

One evening I went into the back drawing-room in which the priest had died. It was a dark rainy evening and there was no sound in the house. Through one of the broken panes I heard the rain impinge upon the earth, the fine incessant needles of water playing in the sodden beds.

At first it appears as though the first five paragraphs act as a sort of "montage" leading up to this particular event. However, the very next paragraph jumps again, this time to when Mangan's sister first speaks to the narrator. This importance of this experience is highlighted by the fact that multiple paragraphs are devoted to it. Though the information gathered thus far gives a good backstory for the narrator's feelings toward Mangan's sister, this is the primary encounter that directly influences the second half of the story. In promising to buy her something from the bazaar, the narrator's "quest" has begun.
From there, the reader experiences a few more time jumps before settling in to the main events of the story; this time, however, the "routine" narrative gives way to specific days and times. From "that evening" to "Saturday morning" to when the narrator "came home to dinner," there is finally a linear timeline for the reader to follow. The final two pages are devoted to the aforementioned "quest," in which the narrator travels to the bazaar.
Though it is not as explicit as some other stories in terms of telling the story outside of a coherent and chronological order, "Araby" definitely has moments in which anachronism is used.
Note: The author's original spelling has been left intact in the above quotes (e.g., "neighbours" instead of "neighbors").