Please don't put heated water into a tank with cold water. A change in more than 2 degrees can be very stressful for your fish. The proper temperature for a betta is 76-80 degrees.
Also, it wasn't clear if you were changing all the water or just a portion of it. If there is no filter you should be changing 25% of the water at least twice a week. If you have a filter once a week should be just fine. Also, please make sure you are adding water conditioner to the new water before putting it in the tank.
Please consider getting a larger tank with a heater and proper filtration. You shouldn't have trouble finding one on craigslist for cheap.
Also, if he is having trouble staying upright, it is usually caused by a swimbladder problem. One reason this occurs is because your fish is swallowing air when it eats. You should try making sure your betta food sinks below the surface of the water before he eats it so that he isn't swallowing any air.
Here's a topic you should read so that we can help you further:
http://www.fishforum.com/viewtopic.php?t=882