As I said before I would keep him in the bowl and give him 100% clean water treated with a conditioner at the same temperature of the water he currently is in.
I would invest in a heater soon with winter being around the corner. Not sure about you, but were I live it gets cold at night quick.
I would read up a lot of betta care so you understand the basics of betta fish keeping. They are wonderful pets full of personality and can live several years when well taken care of.
Here is a sticky on general care. That you should read.
Fish are constantly excreting ammonia through their gills, think of it as urinating. Also when they defecate you have feces in the tank also causing ammonia. After a while it will build up and need removed. Unfortunately fish can not do this themselves and need regular tank cleaning. So we siphon/suck some of it out. Usually 25-30%, and replace it with clean ammonia free water. :)
The smaller the tank the more often you need to do so. A 5 gallon uncycled tank would need at least 50% of the water changed weekly.
Any animal left to sit in their own excrement will end up getting sick.
I would read up on how to cycle your tank so that way 30% water changes weekly would be sufficient.
Here's an article on cycling and the several different methods on how to do so.