Webster Hall is a solid venue, and a frequent stop for many great bands on their rise to glory. If you've only got a few nights in the city, chances are a great act will be playing here.
Occasionally I'll look at the Manhattan skyline and wonder what it was like in 1931 when people around the world were saying, "Holy shit, did you hear what they just built in New York?" Just get up there and trip out about humanity. And be sure to keep an eye out for the unmarked locked door guarding a set of stairs leading to a hidden terrace that was originally designed as a fucking docking station for zeppelins.
While the current trend in noodles may well be ramen, this bafflingly-named Japanese restaurant just ten minutes from my apartment specialises in udon. Unless you are my friend Diego Zambrano it will probably take you several visits to get through all the things you need to try on the menu - the gyoza are otherworldly, the sushi 'tacos' clever enough without being silly. New York has thousands of awful Japanese places; this is not one of them. It's super nice to come in on a Sunday night and eat at the bar next to curmudgeonly old guys barking at each other in Japanese.
You've heard about the reading room at the New York Public Library, haven't you? Then I need not go on. Just look at it. And you're allowed to hang around in there!