Welcome to the forums, eli5112. You're going to learn that betta require a bit more care then most petstore staff tell you. A little tiny bowl like that is like you or I living in a closet. Yes, we can, but we're not happy, right? He's not going to be happy in a tiny container. There probably isn't much room to turn around and nowhere for him to hide when he's stressed or scared.
He also needs a heater. Betta are tropical fish and need warm water to thrive. a heater and a thermometer are a good idea, so you can keep his water temperature at or about 78 degrees.
With a tank less then a gallon, you have to change the water every other day. If you can, every day is better! You can get away with every other day, but it'll leave him open for illness. And keep him as calm as possible. Water changes can be stressful, one of the downfalls of a small tank that needs frequent water changes. The reason he is sick now is probably because he has been in poor conditions for months. Betta are very hardy fish, they can handle a lot of stress, but there comes a point when they get too weak to fight. Don't give up on him, though! It'll take some work, but you can help him!
If you're going to keep him in the tiny bowl, make sure he is in the warmest room in the house. Change his water 100% every day! He might be eating too much and bloat could cause him to float funny like he is. Try fasting him. This means not feeding him for 24-48 hours. It won't kill him, don't worry, just let him stay in a warm dark room with clean water and hopefully he'll recover on his own.
If you're able to get a larger tank or can find something safe around the house that is bigger, do it. He needs to be in something that holds at least 2.5 gallons of water, and if you can get something bigger, that's even better! He'll need a heater, too. You shouldn't put a heater in something under 2 gallons unless it's made for something that small. It's really hard to control how warm the water is in a very very small bowl or tank and you might cook your fish without meaning to.
When a betta is warm and in a big tank with lots of clean water, he acts a LOT different then a cold betta in a little bowl. He'll perk up, be much more active and interested in you. Healthy happy betta are always moving! So you and him will both be happier if his conditions are better.
I really hope this helps and I hope your betta is alright. You can ask us anything, we want to help :)
Oh, and we love pictures!