No, he won't be fine without a heater or a filter. It's winter time now, depending on where you are in the world, and he needs a heater, no matter the temperature of your house, or outside. Keep him in a tank that is 5 gallons or more, 2.5 gallons at MINIMUM. If you do not get a filter, your Betta will most likely die of ammonia burns, AKA, swimming and living in its own waste.
You should put water out tonight, GET A FILTER, or else it will not be cycled. You need to put a few drops of the bottled stuff from the pet store - it's expensive, but it speeds up the process of cycling. Cycling the tank means that all of the cycles, AKA nitrate/nitrite, ammonia, and the pH are perfect. Upon getting your fish, you should get test strips from the pet store, and test your water. If your water does not display as okay for your fish, you should not get him.
If you do not get a filter, you're setting yourself up for cleaning the tank every two days. Nobody wants to do this, especially with school, work, and other things in life. Save up $20, and get a filter. As well as a heater! If you do not get a heater, your Betta will get cold, and mostly just sit on the bottom of the tank. If you want him to live a much longer, and happier life, invest a small amount of money in getting a heater and filter.