Either your fish is stressed out a lot or your water quality is very poor. Non-cycled water, from my knowledge, does not cause fin rot. But I already agreed that cycling your tank is good, but not necessary if you carry out with bi-daily partial water changes. Your fish probably has a bacterial infection, so I don't think a cycle tank solves that at this moment. Your fish needs PWC and treatment. One last thing, be sure the water you use to do PWC is dechlorinated and similar if not exact to the tank water. Usually when fish are stress or have poor water conditions, this increases chance of disease such as fin rot. Fish can live in non-cycled tank, but like I said, it needs daily pwc. And when you do pwc, you have to be subtle about causing stress to the fish as his home is small and he will experience an earthquake lol. Good luck and good day.