Sponge filters are good as they provide an area for beneficial bacteria to grow on, and they don't produce much of a current.
I would strongly recommend purchasing a heater. Temperature fluctuations by even a few degrees can be detrimental to your betta's immune system and make him more susceptible to disease.
An example of how temperature changes can cause problems. One of my heaters wasn't plugged in properly overnight, and by the next morning my otherwise healthy fish had fungus growing on his tail. I turned up his heater, kept the water pristine and added some aquarium salt and thankfully he was fine.
It is also important that the smaller the tank, the more stressful any sort of fluctuations whether temperature or water chemistry will be.