If you visit the General Assembly building, you must have a CHP to enter armed, but you may carry either OC or CC. I am not 100% sure if this also applies to the Capital Building and the Governors Mansion (someone please clarify), but I know that the Capital grounds are ok for OC without a permit.
You need to watch out for any buildings that are part of Virginia Commonwealth University (VCU), as they are spread around downtown, and it is illegal to carry on those buildings.
Almost everything else that I can think of follows standard VA law. Parks (local or national) and local government buildings are ok, and any private businesses can make their own rules.
I have personally OC'd all around the Capital amd GA and to the Museum of the Confederacy in downtown Richmond, and I have never had any problems.