Smack. We can only estimate how long it will take anyone. It could be raining every day, you might like to hike long hours or short hours, you might go alta-light or carry extra weight, etc. You could look at some journals and see how long it takes different hikers to cover that distance. I think it is about 150 miles. My estimate is between 12-14 days, hiking about 12-17 miles per day. It is a beautiful section, there are some parts of it that you will be able to make great time, there are some parts that will be tough.
You can get to Hot Springs by the way of Ashville, NC. Not too long of a drive between those two cities. I’m not sure if you are planning to park your car in Hot Springs and section hike or if you are being dropped off in Hot Springs? Being a section hiker I know it is difficult in getting transportation. There are plenty of shuttles available between Hot Springs and Damascus, but I do not know if anyone would want to shuttle the entire distance, that would be about a $200 shuttle.
Give us a little more detail on how you plan to get to Hot Springs and how much time you have for the section hike. Hot Springs, Erwin and Damascus are great places to relax a day or two. Good Luck.
Bilko