It really varies. Normally the author themselves has to come up with the money somehow though. At least where I've been, you can't just bill it to the university. If you have research grants, you can often pay out of those (this is common in the natural sciences). If you have some kind of general research budget you could pay out of that (e.g. I get $2k/yr unrestricted research money from my university... but I'm definitely not going to spend it on an APC!). Some universities do have an internal OA fund you can apply to, but those are often limited. For example, MIT will cover up to $1k per article for papers written by its faculty/staff/students: https://libraries.mit.edu/scholarly/publishing/oa-publishing...
