It's only about 40 miles from Staunton to Charlottesville. A check of the Greyhound Bus website indicates no service in Staunton, so try this link for a list of taxis/limos and contact them to see what options they offer.
That is about 40 miles through rural area and mountains on Interstate 64. I don't think there are any options other than a taxi.
Sorry, I was trying to edit (unsuccessfully) because I noticed ny link did not work. Just google for a list of taxis in Staunton and see what is offered.
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