The tank is fine, certainly upgrading would be nice, but if you can keep up with water changes to keep him healthy, it would certainly make a nice home for him :) With a 1 gallon it's best to do 2 water changes a week, both 100% so there's no chance ammonia can build up and hurt him :)
I do suggest getting a heater though, 70 is very low. The reasonable temp for Betta's is anywhere from 76-82, they are tropical fish and lower temps will hurt him as well.
Most Tetra foods aren't really good, the things you want to watch out for are fillers; i.e. wheat, grains and corns are especially bad. You want things like Fish meal, whole salmon, garlic, shrimp meal etc. I'm a huge advocate for New Life Spectrum Betta Pellets because my fish absolutely love them and they contain garlic which Betta's go crazy for! It's also a very small pellet, probably smaller than your mini pellets lol, so it's just a little easier for them to eat and it doesn't expand as much.
If your store doesn't care NLS (I had to order mine from online as well) you can get Omega One Betta pellets, that small little container costing 3 bucks is going to last you a long time. Those pellets are a tiny bit bigger than NLS but they carry the same nutritional value for your little guy :)
And just so you know, a bigger tank is easier to maintain but if you can't afford it or anything, don't worry about it, just focus on keeping your new little boy healthy