My niece often goes to college parties and drinks from anyone and everyones cup. how likely is it that she can get bacterial meningitis?

Meningitis is not likely in that scenario, but not impossible. Since college students are going to be college students, all you can do is advise against it. However, the Meningitis vaccine is advised for college age students and she should have that, for sure. Good luck.

The main risk of meningococcal meningitis in students isn't from drink sharing, but just being in a crowded party environment; or even just sharing dorms and classrooms. Drink sharing might further raise the risk, but not by much. Your niece should have had the meningococcal vaccine, in which case she is at much lower risk. Even without vaccine, meningococcal outbreaks are rare in students.

More likely than bacterial meningitis from drinking from others' cups is any other infectious disease, especially viral illnesses...COVID-19, mono, colds, and others...also strep (bacterial, not viral).