In AmE, yes. In BrE, no.
Do we need to use the definite article or no article?
After about the third time of being asked to get out of bed, I get up and have some breakfast before I catch the bus to ___ university.
If I were a native speaker of English, I would never shut up. :-)
I am not a teacher.
If it's the university that you attend you would usually not use the article in BrE. The same goes for 'school' if you study there, and 'hospital' if you are a patient there.
If the words 'university', 'school' or 'hospital' are used to designate actual physical places rather than as the names of institutions, you would use the article.
I catch the bus to university/school = the place where I study.
I catch the bus to the university/school = that large complex of buildings on the other side of town.
I spent last week in hospital = I had been admitted as a patient.
I spent last week at the hospital = I was visiting someone/working there.
The same applies to the word 'prison'.