I don't know, I don't think so. I didn't know what fun rot was when I got him. But his tail tips aren't really white anymore from what I can tell, it seems like they turned black idk what that means. But if it seems like he needs it I'll get some medication, I have no idea what I would use for fin rot medication though ??
The only really "Medication" I've ever used is aquarium salt. If your bettas fins have turned blakc, that's bad. That's called 'Black Rot". It can be tricky to spot on your betta, especially if they have dark fins, if it is a mild case, or if the fish is a crown tail. There are, however, tell-tale signs your fish's fins are being eaten by this pesky bacteria.
(This an exact quote from this website...
bettasplendid.weebly.com/fin-rot-101.html)
Aquarium Salt
Now for the "medicine":
The most effective medicine that you can use to cure fin rot at a mild stage is clean water and Aquarium Salt.
AQ Salt is sodium chloride, which heals open wounds and helps de-stress fish. Aquarium Salt is commonly found in many pet stores and is not expensive.
Like with any medicine, AQ salt should not be used for prolonged periods of time or in very high doses. Ten days is the maximum amount of time you should treat a fish with AQ Salt. Prolonged use can cause kidney and liver damage. Also, like any other medicine, the appropriate dose should be used. Do not use AQ salt as a "preventative" medicine at a low dose, as it will eventually become less effective and diseases will become resistant to it. For example, when 1tsp/gallon of water is needed, do not use .5tsp/gallon. Its like taking a half an aspirin when you need three:; it doesn't do anything!
(I suggest you read that hole article, its a lot of good information.)
Put the fish into a clean quarantine tank and do a 100% water change in that tank. Keep the tank dark and at 76-78'F if you can. Float it in the "big" tank to keep it warm. Add in 1tsp/gallon of Aquarium Salt, pre-mixed beforehand into the tank. For slightly more severe rot, use 1.5tsp/gallon of the salt. Do 100% daily water changes each day and re-dose only after the 100% change, for no more than 10 days. After that time, turn off the filter to aid in fin regrowth and up the water changes for a bit.
(Exact copy from the website mentioned above.)
After all of this is over, I strongly suggest you get the 5g out and clean it up really good. No tank is "cursed", so give it a chance. Good luck.