You need him to cycle the tank, or else you will need to keep feeding a source of ammonia. Some people add fish food but it's messy and will require more work and isn't as accurate and you still need to be able to test with numbers. You can very accurately use ACE Pure Ammonia and an eye dropper.. but again you need to test.
All this is going to be more expensive than just getting test kits, which will run you $15-$30. You could also just try increasing water changes to twice weekly 50% with siphon, but if his fins get worse you will have to remove him.
Fin melt is a condition where the fins look kind of stuck together, bunchy or melted a little bit. It is entirely a condition of water quality issues.