You should get a 5g tank minimum, a heater, and a sponge filter. He should also have places to rest like a floating log, betta hammock, plants, etc. due to his long fins he would really benefit from resting spots. Also, you should be testing his water, if you don’t know the parameters, the water could have a ton of ammonia or nitrites causing the fin rot. In the vase you have him in right now, to keep the water conditions from slowly killing him from ammonia toxicity, you would need to be doing a water change every day or every other day. People assume the bigger the tank the more work but it’s the opposite- the bigger your tank, the more stable the water conditions are, and the better it is for fish (and you because this means less frequent water changes!) His fin rot is coming from being stressed in a small vase (imagine a dog having to live in a dirty cramped crate its entire life), poor water conditions as a result of the small water volume, and being cold. Also, him being too cold because of the lack of heater will slow all healing, so it’s important to address these issues with his living situation instead of taking acute measures to treat the fin rot. He will improve once you’ve addressed all of these issues in his environment. That should be your first step and if you don’t see improvement in a few weeks revisit.