Excellent Neapolitan pizza at Keste. Other possibilities: Classic New York coal-oven pizza at Arturo's is not far away, and you can hear some jazz there before you hear more jazz. There's also a branch of Meskerem on MacDougal. I've never been to that branch but have liked their 47th St. branch near 10th Av. very much. If they have tej (mead), get it!
106 W Houston St, New York, NY 10012
124 MacDougal St, New York, NY 10012
Keste Pizza & Vino
271 Bleecker St, New York, NY 10014
My two favorites right near there are Mamoun's Falafel and Hummus Place. Mamoun's doesn't serve alcohol; Hummus Place serves beer & wine; both are quick & cheap. But if there are more than two of you, be warned - these are small places and not the right choice for a group, unless the group enjoys standing on the sidewalk and eating. I also have a soft spot in my heart for Villa Mosconi, which is less quick and less cheap, but does have very serviceable Italian food, and liquor.
We lucked out to see Dave Brubeck there on our last trip to NYC. We were surprised to find the Blue Note is also a full serve restaurant with apps, meals, desserts, etc etc. We did not eat there, as it just seemed weird to eat a meal while watching a jazz legend.
Anyhow, I'm no expert on Manhatten, but Bleeker is just around the corner with plenty of places to eat, sip, and enjoy.