Yes you can.
It's not a smooth process, since that computer will back up all the data and app packages for all your apps, then restore that backup data but it won't have the apps or music that doesn't get backed up as part of the backup process.
It will work, but you might want to transfer purchases / authorize that computer for a more transparent upgrade process. You can't really know it worked if you don't reload the apps after you restore your "backup".
There's nothing preventing you from doing this and then restoring your backup when your upgraded phone gets back to the computer from the last backup. You have to keep track of things and not do work that will get wiped when you load an older backup, but I've done this many, many times and it works well when you understand a bit how backups work.