Alright, your best bet might to be to use the restore from iCloud option, as it will redownload all the apps from your prior device (definitely make sure you're connected to WiFi when you're attempting to do it so that you don't eat through your data plan).
First you'll want to verify that your old iPhone 4 backup is still in iCloud (you can check from your iPhone 5):
- On your Home Screen, tap Settings
- Tap iCloud
- Tap Storage & Backup
- Tap Manage Storage
- Select the device you want to manage
You should see a backup for the iPhone 4 if there is one. If you tap on that backup, it will give details such as when it was last backed up and the backup size. You'll just want to know that the last backup was able to complete. If you have the iPhone 4 backup still, then you're good to go.
If you decide to proceed with the restore (definitely your call and not mine), go all the way back out to the first Settings page.
In Settings, tap:
- General
- Reset
- Erase All Content and Settings
How content is restored to a new device
When you go through Setup Assistant on a new device running iOS 5 or later, choose "Restore from iCloud Backup" and enter your iCloud account and password. You will see the three most recent backups for each of the devices on which you enabled Backup. Choose which backup to restore from.
After your device restarts, your settings and accounts are restored and Backup starts downloading your purchased music, movies, TV shows, apps, books, Camera Roll, and other content.
Note: You may be prompted to enter the passwords for all accounts enabled on your device. You may also be prompted to enter the password for for any iTunes, iBookstore and App Store accounts for which iCloud Backup is attempting to restore content.
A progress bar will appear underneath the apps that are being downloaded. To prioritize the restore of a specific app, tap its icon. Restore will attempt to download the same version of the app that you had installed previously. If that version is no longer available, the latest version of the app will be restored.
Support Page: http://support.apple.com/kb/HT4859?viewlocale=en_US&locale=en_US