Many years ago I attended classes (sporadically) at a tiny college in Maryland whose staff and students consistently referred to it in internal correspondence as "The College." The college's quarterly magazine was even named The College. Then I attended classes in graduate school (even more sporadically) at a gigantic state university that so dominated its in-state peers (in the minds of its staff and students) that many people would buy stick-on decals for their car that read "The University of Texas" and then snip off the "of Texas," confident that everyone would know what school "The University" referred to.
In both instances, however, the definite article was always included with "College" or "University." Without "The" (or at the very least "the"), the designation would have become too amorphous to sustain its sense, I think. In any case, I would spell university with a lowercase u in the example sentence you give, since I don't see any improvement in meaning to be got from the capital U.