Basically your problems are
a) Bettas are tropical fish (they prefer a water temp of about 80) and don't do well without a heater unless you live in the Sahara.
b) The larger the tank, the more diluted his waste will be when it dissolves in the water. So in a really small tank, his dissolved waste (aka ammonia) is highly concentrated. You may recognise ammonia as the active ingredient in commercial cleaning solutions. In a 1g tank, you would have to change the water 100% tevery 2-3 days o keep ammonia levels safe for fish. Anything below that you will have to clean once a day or even twice a day. You are better off picking up a 5g tank from petco that only has to be cleaned once a week. Always remember: the solution to pollution is dilution!
Here is an awesome care sheet on bettas. You will be suprised at the amount of care they require compared to what you thought they needed based on the many myths circulating about bettas.
And a mythbuster about the natural habitat of bettas. Hint: A bowl will not
keep them happier than a full-sized tank!
By the way... welcome to the forum!!!