Well, obviously you are into music since you're on head-fi. Do you like jazz? If so I highly recommend going to a jazz club such as Village Vanguard, Birdland, Blue Note, Iridium, etc. If you go to the Village Vanguard, just around the corner is Small's jazz club which goes all night until like 5am. The U.S. is the birthplace of this art form and NY has the most extensive jazz scene in the country, not to be missed imo. Here's some links:
The Village Vanguard Web SiteBlue Note Jazz Club New YorkIridium Jazz Club New York City - NYC ManhattanBirdland, The Jazz Corner of the World!Jazz Clubs in New York City (link to all NY jazz clubs)
Wow, Mike Stern is playing at the Iridium from Dec. 23-27 plus three other greats...a tribute to Miles. This would be a fantastic show.
For Broadway, there's a ticket booth in the middle of Time's Square that sells day of show half price tickets to just about everything.
When I lived in Brooklyn I was pretty much broke so a favorite pastime was just walking the different Manhattan neighborhoods like So-Ho, The Village, Chinatown, Little Italy, East Village, etc. Brooklyn Heights is just across the Brooklyn Bridge. It's a beautiful neighborhood with nice restaurants and a promenade overlooking Manhattan and the NY harbor.
I'm not a big fan of the usual tourist attractions like the statue of liberty, but to each his own.
You won't need a bus pass. Just get a city map, and the Subway will take you anywhere you want to go. You can pick up a Subway map at any station. At Battery Park you can catch a ferry to Ellis Island or the Statue of Liberty, or a free ferry to and from Staten Island if you just want the views of the city off the water.
You can do some serious damage in a week if you do your research and study the map.
Hope this helps. Have fun!