does the university typically cover these costs or are they borne by the authors of the paper?
This seems to be a false dichotomy — usually it is not an "either-or" (unless you submitted to an open access journal with a mandatory Open Access fee).
It might be possible that your university does not cover the Open Access fees of this journal. However, that doesn't mean you as authors have to or should pay. Many universities subscribe to journals in order to make sure their researchers and students have access to papers which are not Open Access.
In other words, even if you don't pay the Open Access fee, many if not most researchers will still have access to your work. (Looking at the journal page on Elsevier you can look at the list of Open Access papers in that journal — often it's a minority.)
In fact, you are probably allowed to "self-archive", i.e. upload a preprint to the arxiv, for example. That way your article is available to everyone freely (open access). Again, you are probably even allowed to update the content of the preprint to match the content of the published version.
Check with your journal for their policy on self-archiving — the database http://www.sherpa.ac.uk/romeo/search.php is also a useful resource.