Hey! I like my big eater betta
First off, I feel the 0.5 is too small. Try finding a 2.5 gallon, since it is easier to heat. That's right - heat! Betta fish are tropical. He will perk up and look healthier once he has a heater. I recommend an adjustable heater, to get the water to a stable temperature of 78-80 degrees Fahrenheit.
Right now, you need to clean the tank every day. Ammonia is lethal to fish... Which is why a 0.5 won't do you any good especially if you become busy and are unable to clean every day.
Food: pellets are better than flakes. Less messy, more nutrients, and does not cause SBD (swim bladder disorder) as much. And easier to count how many you are feeding. Try finding some variety for your fella. I feed my bettas thawed bloodworms, which you can find in pet or fish stores in your area.
You don't really need a filter, as some bettas are ushed around by it and can become stressed. If you get a 5 gallon, I would say a filter would be fine if you can baffle it... Plus then you can cycle the tank, making it easier on you.
Hardy live plants: moss ball, java fern, Anubias, duck weed... Otherwise go for silk and fabric plants since they are less likely to snag your fish's fins, like plastic can.
Hope that kinda helped ^_^