The three act plot analysis best applies only to Part II of Notes from the Underground, since this is when we have action instead of philosophizing. You can see the elements of the three-act structure most clearly in the story about Zverkov's dinner. Act I lasts up until the Underground Man invites himself to the dinner and then obstinately declares to his reader that, no matter what, he is going. He's set up the dominoes, and there's no turning back now.
And it's only downhill from there. Matters continue to get worse and worse all through dinner as the Underground Man gets further and further from ever being accepted by Simonov and his friends. We think the all-time low-point is when the Underground Man begs Simonov to lend him even more money so he can tag along to the brothel, even though he is very clearly not wanted.
Act III takes us all the way through the Underground Man's tryst with Liza. He resolves the Simonov issue by sending an apology letter, but we're pretty sure that's isn't going to do the trick. He's likely to be a social outcast for the rest of eternity, or at least until he dies. When Liza slams the door and leaves the Underground Man alone, we're in conclusion land; the Underground Man is right back in the inert stagnancy he started in.