Okay. I kept my betta in one of those one gallon glass fish-bowls too (cleaned once weekly) and they do go down pretty slowly in that kind of housing.
Say hello to your betta bible:
You will want to get a heater. Bettas are warm-water fish and need a heater to survive long-term. Since there are no heaters which can heat a 1g-or-under tank without boiling it, you will need to get a bigger home for your betta. The average 'minimum' most betta keepers reccomend is 2g, but alot of people even insist on 5g for a single betta!
In order to circulate your water (just a little, mind you!) you will need a filter. Moves water past the heater, distributing the heat evenly. Also, it reduced the frequency with which you need to clean your tank and supplies your betta with extra oxygen (they can and do breathe through their gills, as well as through their labyrinth organ).
If, hypothetically, you managed to keep your betta in a heated 1g (ie by heating the whole room) you would need to change the water every 2-3 days to keep the ammonia manageable. So water changes are probably a good place to start. I keep my guy in a 5.5g and can even get away with cleaning it every 2 weeks (which I am to an@l to do lol) because of all the space, plants and filtration to keep the water fresh.
Finding an all-in-one kit like the mini-bow 2.5, etc is an economical place to start. Good luck!