Do you know how you'd get from Boone to Grandfather mt? That's only about a 25 minute drive, but even that will be a problem. I do remember seeing shuttles to/from Blowing Rock when we were at the Highland games a few years ago, but I think I remember that had something to do with a road closure, so you'd need to double check if that's still going to be available.
As for getting to an airport, I know 2 private bus services run routes to/from Charlotte during the Appalachian State University (in Boone) regular academic year, but am unsure if those routes run during the summer months. Here's links to one, the Hickory Hop, you'll need to contact them to find out if routes will be running while you're there. http://www.hickoryhop.com/BooneSchedule.html
Also, here's a reply I wrote recently on another topic in Blowing Rock forums, wanting info on public transportation, etc in the area and to Charlotte, the Horizon coach route is the other bus service to/from Charlotte......"Here's the only options I know of.. admissions.appstate.edu/visit/transportation The Horizon coach routes listed are mainly set up for ASU students/community, but others can and do use it... (BTW, from CLT to Boone is less than 2 hours drive by car; the bus services/shuttles, etc. take almost twice as long....that is when they're running on time)
Don't know what your plans in Boone/Blowing Rock...but if you manage to get to Boone by public transportation, you can use the "Appalcart" (Boone bus service run by the university, but public can use) to get around Boone, but it doesn't go to Blowing Rock, or anywhere much outside Boone town limits. You'll be severely limited relying on public transportation to get to the area, and also while in the area.
For a number of reasons, I'd rent a car."
The "Appalcart" in Boone mentioned above, has limited/fewer routes during Summer months.
Best of Luck :)