Zebra, you seem like you are really trying, and that's good. A lot of the people here are very passionate about fish and really want the best for all of them, even yours. Get the pellets, I use Hakari Bio-Gold, but New Life Spectrum is really good, too. And a net is best in a small container like that, or, if you're going to use a cup, use a clear one, so he won't see it coming, less stress. The net won't hurt his fins and as long as it's done quickly without any accidents, moving him either way is fine. Just get it done fast so he's not too upset. (Thanks for the wording, I had a mental image of a ceremony centered around moving a betta from one container to another.. long black robes, chanting.. yeah... I'm strange.)
Do you have a thermometer in there? If you said you did, I somehow missed it. Grab one when you have the chance. they're only a dollar or two. All of this will be a great start to getting him healthy and happy. But as soon as you have the money for a larger tank, get one. He'll be a thousand times happier and healthy in something 2.5 gallons or more. even doubling his current home would be good. I know sometimes money is tight, but thrift stores sometimes have bowls and containers that are large enough for a betta, and you can check Craigslist. Or ask your friends, someone might have a tank they're not using.