Georgetown operates the GUTS shuttle beween the campus and the Rosslyn Metro station (Blue and Orange Lines) and the Dupont Circle station (Red Line) all day long, and particualrly during rush hour. You can check out the schedules (and other info) here...
Georgetown University Transportation Shuttle (GUTS)
The shuttle is quick, convenient, and FREE. With that, you can pick a place to live pretty much anywhere and still be able to commute into Georgetown reasonably easily. $2K per month will get you quite satisfactory living space almost anywhere. The short-term thing could be a bit of a problem, as many places do want a 12-month commitment, but many don't.