Dude that will take most of your day to do that. I tried to make the drive from Asheville down to the Dragon to make that drive, but I overestimated how quickly I could do it and had to turn around before I got there in order to make it to a more important event in Asheville.
First off, DONT SPEED on the Dragon. Second off, the road will be heavily traveled as it is a very popular spot for all enthuiasts, both motorcycles and car enthusiasts. Third, pay attention to the bikers if they are making hand signals at you. If you ignore them and round the bend, its very likely you could be run-off the road by a 18 wheeler who didn't know any better that they were trying to escort thru the Dragon.
Also, Graham County, NC cops run radar on US 19 heading down there. NC Highway Patrol likes to run radar on the roads around there by sitting at the bottom of the mountains waiting for people to coast down the hill and thus speed in the process. If you have a radar detector you should be able to pick them up.
Fastest way out of that area to I-40 is by taking US 19 North, and then pick up the Great Smoky Mountain Expressway which will take you to I-40 just west of Asheville. Its about a 80-90 mile run I think between the Dragon and Asheville.
As for other cool things to do that are somewhat by the Dragon: Harrahs has a casino in Cherokee, NC, which is on US 19 north of the Dragon. Cherokee is also a bit of a tourist hangout with some other attractions. If you're willing to make the drive into TN, Pigeon Forge is a HUGE tourist place with a lot of tourist attractions.
Nothing is close though down there, so keep that in mind. Don't expect to tackle the Dragon and then expect to make the drive to VA in the same day.
Website for the Dragon is:
http://www.tailofthedragon.com
Be sure to have fun.