They do need a heater, there water temp needs to stay somewhere from 76-82, but don't let it fluctuate.
You should get a bigger tank, at least 2.5-3 gallons. A filter really helps keep your water clean, but if you don't, use live plants and do 100 percent water changes a couple times a week (or more? I'm am not sure sense I use filters...) With a filter, you can do cycle the tank first before adding him, it's better that way. I already had my betta before I cycled the tank, so I am going to use Prime (water conditioner that helps slime coat and detoxifies water during cycling to keep your fish safer) every day with partial water changes.
Also, getting a small thermometer will help keep track of the temperature, make sure you use water conditioner and keep the temperatures the same when changing the water. And a light, you can get an aquarium kit that comes with a filter and light, flourescent is best because it doesn't heat your water up and you can see things better. :)