No, salt is not the answer to ammonia (that's for nitrite) but water changes are. Do 50% or more daily. You really need to have your water tested, if you don't have a kit, go to the lfs at a minimum of 1/week, more is better.
Putting fish in an uncycled tank is risky even with lots of water changes. Fin damages is only part of it, gills can be harmed too...& maybe other parts. I hope you're using a dechlorinator like Prime that "binds" ammonia for 24-48 hours. Not all do.
He's a pretty betta, you'll need to work hard to make sure he's healthy for the long term.