Okay it sounds like you have been the vitim of pet-store misinformation. You will soon learn that most employees of pet stores have been trained with the intention of turning the highest profit for the store, not necessarily informing the owner. Good marketing does not equal good advice!
I suggest you read our Bible
as a starting point.
Here is a bit of information on why a bowl will not cut it for a betta:
Unfortunately fish stores will sell you just about anything to get your money. And they make considerably more selling cheap tanks to a bunch of people than they do selling proper tanks to the few people who have actually done their homework on tank size.
Another common myth is that small tanks are lower maintenance. If you put three drops of red food coloring in your 1g bowl, it would probably turn the water bright red. If you put 3 drops of red food coloring in a 5g tank, it would only turn the water a light pink. The same goes for the waste your fish produces. The smaller the tank, the stronger the waste will be. So you have to clean it more often. If you are looking for a truly good beginner size, I strongly recommend you grab a 5g tank.
Furthermore, there are no heaters on the market that will heat a 1g to the right temperature. Since it is such a small amount of water even the smallest of heaters will often make the water too hot.
As mentioned above, you will probably want to go with a 5g if in addition to a pet fish you also have a pet 3 year old in your care. Maintenance will be far easier and quicker in a larger tank, and you have a lot of margin for error should something go off with the water quality, etc.
I suspect your pet store recommended feeding every three days because if your betta is living in an unheated bowl where his metabolism is frigid and he has no room to exercise, underfeeding the fish is a good way to reduce the waste it produces and thus exend it's life in such a small enclosure. This was common practice originating with the invention of the goldfish bowl years ago in China. When fed sparsely and kept below optimal temperatures, the fish would survive longer indoors. However while sparing feedings are a way of getting your fish to survive for a few years, he is by no means thriving which should be your goal as a fishkeeper.
Now onto your betta's actual health problem. It sounds like he has an infection due to scraping himself on the (plastic, I assume) plant. The rough edges found on plastic plants are likely to scrape or tear a betta's delicate fins... in fact, if it will tear a nylon stocking it will probably do the same to your betta! You may want to sand the plant's edges a bit and buy only live or silk plants in the future.
You will want to change the water daily and add aquarium salt to discourage bacterial growth until he gets better. Keep a close eye on it... if it spreads despite aquarium salt and clean water we have a problem. As I mentioned earlier, a heated 5g would make for an optimal recovery. Bacterial infections can become very aggressive if not nipped in the bud.
I know that was a lot of information but I hope it helped you a little!
Good luck and welcome to the forum!