In A Midsummer Night's Dream, how is Hippolyta's reasoning concerning how quickly the next four days will pass different from that of Theseus?

Great question! At the beginning of William Shakespeare's comedy A Midsummer Night’s Dream, Hippolyta and Theseus are due to be married in four days. The upcoming royal wedding serves as the backdrop for the plot of the play, and most of the characters are involved with the wedding in some capacity.
In the opening lines of the play, Theseus bemoans that time seems to pass so slowly. He says that the moon "lingers my desires" and essentially says that he wishes to bed Hippolyta immediately rather than waiting until after the ceremony.
Hippolyta is much less anxious. She responds to Theseus's impatience by saying:

Four nights will quickly dream away the time

For Hippolyta, time is moving quickly. She refers to the night of the wedding as the "night of our solemnities." By using the word "solemnities" to describe a (normally) happy and joyful experience, Hippolyta reveals that she is not enthusiastic about her coming marriage.
For Theseus, four days seem like an eternity while for Hippolyta four nights will pass far too quickly. This is interesting, and the odd relationship between these two characters is revealed a few lines later when Theseus tells his betrothed:

Hippolyta, I woo'd thee with my sword,And won thy love, doing thee injuries;But I will wed thee in another key,With pomp, with triumph and with revelling.

This reveals that Hippolyta is not marrying Theseus willingly. The Queen of the Amazons has been kidnapped by Theseus and is being forced into marriage. This detail largely explains why Theseus and Hippolyta have different perspectives on the passage of time leading to the wedding.
I hope this helps!