These events take place during "carnival season" in Italy, the two or three weeks just prior to Lent, the forty days leading up to the Easter holiday. This is why there are so many parties and lots of people out and about, drinking and carousing into the wee hours.
Montresor meets up with Fortunato, a man he considers to be his enemy, in the streets at dusk. Fortunato is dressed in costume because of the holiday season, and Montresor has carefully chosen his own costume to disguise his face and body so that no one will be able to identify him later. By manipulating Fortunato, Montresor is able to entice his enemy into his family catacombs beneath his home. Much of the story does take place there, as this is where Montresor leads Fortunato far into and underneath the earth, eventually trapping him in the wall and sealing him up there to die.
At the end of the story, Montresor reveals that it has been "half of a century"—a full fifty years!—since he killed Fortunato. It is unclear exactly where he is now, but as we can imagine that he is an old man now and that he has never confessed his sin before, I think that he is confessing his sins to a priest because Montresor on his deathbed. In the story's first paragraph, Montresor had said that the person he is speaking to "well know[s] the nature of [his] soul," and so I think a plausible interpretation is that he is speaking to a priest to whom he has confessed other sins before; such a priest would understand Montresor's character. The final setting, with this interpretation, is Montresor's deathbed, perhaps resting in his own home, his own bedchamber.