if "good new york experience" suggests you are coming from out of town or lunching with people from out of town, i'd suggest dining in a different neighborhood, as i think the UWS has incredibly weak/lame restaurants overall...
when i am around there, there are a few places that i go to: Nice Matin (good tuna tartar), Cafe Ronda (Argentine/Spanish tapas, sandwiches), etc...but there isn't a single restaurant in the whole UWS that i'd travel to specifically for either food or atmosphere...but if you're looking for a place of convenience, Nice Matin or Cafe Ronda could work, but neither is especially NY in vibe...
201 West 79th Street, New York, NY 10024
249 Columbus Ave, New York, NY 10023
If you want casual, I'd recommend Good Enough to Eat, though it's over a mile from Lincoln Center. If you want a higher art experience, I definitely concur with the Telepan recommendation, though it's somewhat over your price point. But it is quite close to Lincoln Center.
Good Enough to Eat
483 Amsterdam Ave, New York, NY 10024
72 W. 69th Street, New York, NY 10023
Recipe (very casual)
Ouest (business casual)
B Cafe West (casual)
If you want very inexpensive, look into Maoz (actually an int'l chain, but very NYC as far as I'm concerned) or Absolute Bagels, or pick up a knish from the prepared food counter in Zabar's (inside the store, not the cafe; the cafe knish are never as good for some reason).
2245 Broadway, New York, NY 10024
2315 Broadway, New York, NY 10024
2788 Broadway, New York, NY 10025
2047 Broadway, New York, NY 10023
566 Amsterdam Ave, New York, NY 10024
452 Amsterdam Ave, New York, NY 10024