iPhone 4 signing in issues
The best way to solve this is to call Apple Care (http://support.apple.com/en-us/HT201232). They would be able to solve this if you want to download an app.

When you want to download a free app, you could also try on iTunes for a Mac of windows pc. When you're asked to enter your payment credentials, there should be an option to choose "none". I would try this first before calling Apple Care.