I live "in" Charlotte & I gotta say you picked a great time to be "stuck" in NC, especially being this close to the mountains. I'm new to this forum (actually, this is my first post) but I am not new to riding in WNC (or coastal but I would suggest heading west). Definitely check out the links that the other riders previously posted as they are great suggestions. If you only have 1 day you can easily hit the Blue Ridge via Chimney Rock/Lake Lure (have a tasty lunch with a fabulous view at Larkin's on the Lake)...find the back-roads to Hendersonville or Black Mountain...hike Craggy Gardens off the BRP or the summit at Mt Mitchell...and still be home by midnight. If you've got an extra day head for The Smokies - Nantahala, Cherohala Skyway, Deal's Gap for Tail of the Dragon...I could go on forever!
If you need suggestions for rentals try Carolina Harley in Gastonia (they are closing their doors sometime this month but, if they still rent, the location is ideal), next I'd say Pat Roger's, or Speedway HD, in Concord (because that's where I bought my bike & I love the way they treat their customers), and I think the dealer in Matthews has the best rental rates but less convenient to the mountains.
Whatever you decide, the best of NC is far beyond the city walls! I would definitely rent yourself a bike & experience it!
