YES THAT IS FIN ROT. Defiantly sounds like Fungal bacterial fin rot Dx Get either Jungle Fungus Eliminator or Jungle Fungus Clear. Treating him to some salt baths
should do him some good too.
For a tank that small you need to do water changes once every 1-2 days! Since it's probably not cycled, the build up ammonia in the tank caused by fish wastes and rotting food the betta didn't eat is deadly and stresses your fish out, making it more likely to get sick. He should also be kept at around 80 degrees F.
Once the fin rot is gone, treat him with some bettafix as well Dx it'll help with healing his fins. If his behavior doesn't improve after that, he might have an internal infection.
Also, once he's all better, everyone here will really recommend you upgrade him to at least
a 2.5 gallon with the whole set up (filter and heater). He will simply thrive in that environment, becoming a beautiful happy betta and that you can admire proudly!
:) Plus larger tank + filter= requires less maintenance (two 25% water changes every week should do for a 2.5gal. Bigger tanks will require water changes less often still!) = less stressing your fish out, since frequent water changes will more or less stress your fish out as well, once again making him more likely to get sick...
It may sounds like a lot of work and money, but it's very rewarding seeing your fishy friend very happy!