mik5,
When I daven mincha at my office, if possible, I will block out a 15 minute period on the calendar for a small and private meeting room. If I don’t have a reservation, I opt to daven in the break room rather than my cubicle, because people may stop by my cubicle with questions, raising the chance of interruption.
In your case, the lounge might be the best bet, even though it’s probably not a quiet place. If the lounge does not work, I’d speak to a friendly faculty or staff member, or even a security guard. More often than not they are delighted to help, and their solution might be much better than a stairwell. At my first job location, my start time was too early to make shacharis with a minyan. I spoke to a colleague who managed the office IT equipment, and he pointed me to an IT storage room that almost nobody went into except for him. He jokingly called it “the synagogue” after that.