What about a picture? Pictures can sometimes really help those are here you know what they are talking about help us.
Some stuff you are going to find out is that you betta needs a bigger home. 1/2 gallon is WAY to small, I still have 2 of these bowls myself but the sit in the closet now. The betta can live in it, but it won't be happy or as healthy as it can be. Most people on here suggest a minimum of 2.5 gallons per betta. He also needs heat. Bettas are from Thailand and China and used to higher temps. It's best for your betta to be kept around 80 degrees. In smaller tanks a filter isn't needed, but frequent water changes are.
As to what is wrong with you guy. Have you checked
to see if it's covered in there? Have you seen ick before on a fish? I don't know if it can happen as quickly as you say you betta got sick, but it's the first thing that comes to my mind when I think of a cottony white sustance.
Whatever it is, you can be sure that keeping the water clean will help. So my suggestion is to make sure to change the water daily. Other than that I don't know enough about betta heath to feel comfortable telling you how to treat him. I hope someone can help you soon & you guy turns out fine. Let us know how it goes.