Ithaca has few choices.
Cornell has a small room with 6 tables. It's poorly lit and they often use it for storage, especially over the summer so sometimes there are only four tables available... not to worry, though, it's usually empty.
Ithaca College has a "game room" with 5 or 6 Gold Crowns. It's better lit and free even for non-students.
Uncle Joe's Sports Bar has three barboxes. This is the main location for APA and NAPA leagues.
The Rhine House bar has two squirrelly barboxes and free pool on Sundays.
The 77 Club is in Cortland, approx 30 minutes out of Ithaca.
Your other options are Binghamton, Elmira, or Syracuse, all ~60 minutes away (though Elmira's about 40).