Connect you old iPhone 4 to your computer and open iTunes, then under File>Devices>Transfer purchases. After that connect your iPhone 5 and select your devices name. Under the tabs at the top of the page (Summery, info, music, apps, ect...) Click on apps you will then be able to sync all of you apps. Follow these articles they will help out a lot.
Plug in your iPhone 4.
Open iTunes and Backup the iPhone.
After it completes, plug in your new iPhone.
Click Restore from backup and select the iPhone 4 backup you just made.
(note the two posts above will only sync apps, not the data from your current iPhone)