i would move to a little larger tank. i have an eclipse 3 gallon, and it suits my betta nicely - without being bulky or big. you will not only being providing more swimming space, but also nice filtration.
and yes, you only change a portion of the water- and try to keep it around the same temp as the old water. as for the amount of water....if he is living in a half gallon, you might need to change most of it out. the less water, the more easily waste will build up.
feeding once or twice a day is sufficuent, just a few pellets or what he / she will eat in about 1-2 minutes. you should also look into feeding a variety of foods- flakes, pellets, brine shrimp, peas etc.