Fin Rot in and of itself is not contagious, but if you keep fish with another betta you may have it catching the rot. This is because the bacteria that cause Fin Rot are opportunistic bacteria - poor water conditions stress fish out, and when fish become stressed, their immune systems become weak. Weak fish are not able to fight off illnesses that healthy fish easily can. So if you have poor water conditions or something else causing stress in the occupants of your tank, you may have an "outbreak" of Fin Rot. From what I read, your fish does NOT seem to be in poor conditions, so when you got him did he have it? Or is he just stressed?
SO start off with isolating your fish.
It is much easier to keep the water absolutely pristine when you have a fish in an isolation container. The secondary big reason (maybe this should be first) is that many medications used to treat fin rot can kill your biological filtration, live plants, and may be harmful to other animals in your tank (depending on the medication and the other animals). It's also just generally a good idea to isolate any sick fish when treating them so that they are not getting picked on and there is no chance of passing any secondary infections on to other fish, as well as making it easier on you to monitor the health and progress of your fish
You may have good luck getting rid of it by simply increasing the number of water changes you do. If your betta is in a small, unfiltered tank, simply doing daily 100% water changes for a week may reverse the beginning stages of the illness. Adding 1tsp of aquarium/rock salt per gallon of water to the isolation tank along with doing daily water changes may also help take care of these bacteria in the very early stages.
Some good medications to treat fin rot are Jungle Fungus Eliminator and Tetracycline. If you are unable to find either of these medications, other options are Maracyn II (you can use it in combination with Maracyn if you want to cover all the bases) and Kanacyn. Simply follow the dosing directions on the medication, keeping in mind that since you are treating in a small container you will have to do full water changes and add new medication instead of doing the partial changes many medications specify.
Hopefully, your fish will recover(which im sure he will) and live to a ripe old age, lol.
**EDIT: ALSO I use Stress Coat. It'll help the fin regrowth and establish a protective layer that will prevent further complications.
*****EDIT 2: Relized UrbanHavok already mentained StressCoat. Sorry I didn't read any of the further posts