Don't worry. It likely won't last long.
Before you say I'm being unkind, here's a basic breakdown of what is happening.
The betta produces waste. The waste goes into the water. In a bowl, there is NOTHING to process that waste into a non-toxic form. The waste accumulates to the point that it is harmful to the fish. Bettas are more tolerant of horrible water conditions because unlike most fish, they have a special organ that allows them to breath air rather than extract oxygen from the water via gills. That's usually pretty lethal to fish, have their gills burn by toxic levels of ammonia. With bettas, they can keep breathing despite the toxic chemicals (ammonia, primarily), which irritates the skin and eyes, making the fish sick and prone to infections. Since the bowl is unheated, the fish will be lethargic anyway, and have a lowered immune response, making it less able to resist infections and heal. It will eventually kill the fish.
All that's happening in your bowl. Right now.
Now, with a regular tank, and a filter, and some care on your part, the tank can be cycled in advance, and the filter will host bacteria colonies that breakdown that waste into non-toxic forms. A heater will maintain the temperature in a range that's good for the fish as well as the bacteria.
Healthier for the fish. Happier for the fish.
If you want that bowl, then start doing water changes of 50% per day. Get the tank someplace where the temperature will be appropriate--and that means mid to upper 70's. Every day, clean out any solid waste and uneaten food. Do the water changes with dechlorinated water. I'd suggest a test-kit as well.