As a non-student, you are fine on UVA property. I can't promise you, of course, that you won't be hassled a bit. But you certainly can't lawfullybe arrested or detained solely for open carrying, and only recently the Attorney General issued an opinion noting that public colleges and universities in Virginia do not have the authority to prohibit licensed concealed carry nor open carry (with only a few, narrow, slightly ambiguous exceptions - none of which could be construed by even antis to include simply walking around the gorgeous campus).
Someone else will have to share their wisdom as for Monticello.
I have walked around the JMU campus openly carrying a .357 without a problem, a couple times now.