If you're a beer drinker at all, you have to check out McSorleys Old Ale House. It's small, its packed, but its VERY cool for a quick beer (Only can order "light" or "dark"):
McSorley's Old Ale House - Wikipedia, the free encyclopedia
Staying on with a bar/museum, Fraunces Tavern is a bar (maybe my favorite I've been to)/musem:
Fraunces Tavern® Museum
You don't need to drink to enjoy either of these two places, both very cool with a lot of history.