Virginia and Ohio share reciprosity agreements; you might want to review these areas first for CC'ing here. As for open carry, there are no laws against this so it is the default mode of carry here. Since you have an Ohio permit, I'm sure you can CC, or OC if that is your wish, in your car or on your person. As for establishments with ABC-ON (alcohol served for consumption on the premises), you must OC there if you are to carry at all.
Businesses may post as they see fit and whether or not you chose to honor their wishes is your decision to take. Since they have no power in law, you can only be asked to leave, which you should immediately in order to avoid any chance of being arrested for trespassing. If you are concealed, then who would know? You decision on this one.
As for UVA grounds, as I understand the laws, you may carry there open or concealed since you are not a student. I would suggest CC'ing just to avoid any stops or hassles.
Hopefully someone from C-ville will soon chime in and fill in the blanks and/or correct my statements where needed and if needed.
Hope you enjoy your stay and return to visit other parts of our fine state.