Ah, I was fooled once again by the et at the end of Universitet.
Mot instead of till?
It works as a sentence, but there's a slight difference - it would mean "I run towards the university." Of course, that's often the same thing - but not always.
What about running the university?
The closest is probably driva, although that only really works for companies and Swedish universities aren't private. So there's no exact equivalent.