My end goal is to become a professor of either mathematics or philosophy. I haven't made up my mind which. Thus, I plan to get a master's degree in math, specializing in logic, in order to continue my education without committing to either discipline. More generally, I suppose my question is: is this a sensible plan, and is the University of Vienna a good place to carry it out?
Kurt Gödel is an alumnus of University of Vienna, and would be a pretty hard act to follow, but at the intersection of math and philosophy he has few peers.
I think your plan is perfectly reasonable, though if you don't speak and write German, you might want to address that. Maybe not essential today, I guess.
The Wikipedia entry on the university is informative, of course.