What do i need to upgrade a Mac? I'm a PhD student and I work with a Mac which is not mine (I mean, it's property of the university). I'd like to upgrade the OS to the most recent version (compatibly with the hardware and other stuff...), but I don't now how this works.
The questions are the following:


Is the Mac "connected" to an account? (Actually I use it with no problem, but ...)
To upgrade the OS, do I need to use an Apple account? (I think yes.) Does this account have to be associated with some purchase done in the past?
Can I downgrade if something goes wrong?
I'm not sure what you mean. You need to be more specific. If it's that the Mac is tied to an Apple ID, no: only the current installation and user accounts are.
Yes, you'll need one to download the installer unless you get it some other way (someone else downloads it, you use internet recovery, etc.)
Sort of. To my knowledge, you can only do so if you have a Time Machine backup of relevant files from before your upgrade. You'd be able to move most changes back over once you finished the downgrade, but I don't know of any easy way to restore an older version directly from the current one without losing data.