Q: Transferring iPhone 4 to iPhone 5

My iPhone 5 broke, its still semi-usable, I just purchased an iPhone 5, unlocked. What worries me is after getting my original iPhone, and having a bad experience with iTunes wiping my phone again when I hooked it up to iTunes after already setting it up and having data on it, I have not backed it up since. Neither is it backed up or connected to the iCloud. A cellphone help and repair service in my area told me, that they could back the old one's data to the new one regardless of the circumstances, but their hours are few and busied with other things. How can I get all my data, notes and applications from my barely usable iPhone 4 to this iPhone 5, with out ever previously backing it up?