I expect you'd be able to find metered on-street parking. Note that the city of Cambridge meters only take quarters (no dimes or nickels). The Meadhall is a couple blocks from the east end of the MIT campus. Here is a list of parking garages in the area if you can't find a metered street spot: http://web.mit.edu/facilities/transportation/parking/visitor...

Hopefully someone else can chime in here, I sold my car last year in favor of public transportation. But there is a commuter train from Providence to Boston and it should arrive in South Station which is connected to the red line and only 3-5 stops away from south station.

It may actually be cheaper/faster than driving. I don't know what time trains come back though. You will need to check on that so you don't get stuck in Boston.