Well, there are great bars all over NY, but the closer you get to Times Square, the more packed they will be on New Years Eve. Also, you should realize that places all over town will presell tickets to their New Years events. Just wait until about early November, and go to a site run by a NY paper (like the NY Post or the NY times or similar), and you'll see tons of ads for New Years Eve and for broadway shows for tourists.
Either that--or pick a place you might like--maybe a jazz club on a place across town--and call them and ask if they will be open New Years and are selling packages. Then leave about 11:00. If you want to go cheap, find someplace not having a party and just hang out until late, then go.
If you can, try to get a cab across town or take the subway to somewhere close to Times Square (which is at 42nd and Broadway)--best viewing of the ball drop is around 44th and Broadway--but it will be fairly packed. If that's not possible (and it may not be, then you'll have to walk over from wherever you are--or just stay where you are and enjoy the New Years celebration there).
Best place to see the Times Square ball drop event is from the Manhattan Marriott--but they have fancy parties and sell out months in advance and they are not cheap. Otherwise, do like I said above.
Enjoy