How to backup data for Shazam app from iPhone 3GS
I think there is no way to achieve this without jailbreaking the device.

I've just checked the Apple developer tools, but they only allow you to get the content of apps that you developed yourself, not apps downloaded from the AppStore. And even with these you can only backup, but not restore :-(

I'm pretty sure it's possible on a jailbroken phone, but I have no expertise in that field, so I can't tell how.