Is there a way to fix my billing problem without having to pay for the purchase that has caused this problem in the first place? Please help

Purchases are considered final but you can try the 'report a problem' page to contact iTunes Support and see if they will cancel the purchase : http://reportaproblem.apple.com

Or you can try contacting iTunes Support via this page :http://www.apple.com/support/itunes/contact/

If they won't you will have to pay what you owe before you will be able to download anything else, including redownloading past purchases and updates to apps.