apps aren't part of the backup, only the app data.
on your phone go to the appstore - update - purchase - all or not on this phone - hit the cloud next to the app to download it back to yoru phone.
If you want to restore from your icloud back up then look here http://support.apple.com/kb/HT1766
If when you fired up the new 5S you provided your iCloud account info, and if you selected an iCloud backup to install, then the Apps which are defined by your backup will start downloading. The Apps aren't part of your backup, but their inclusion on your iPhone is and they should start populating if you indeed backed up to iCloud and selected your iCloud backup.
In any case, you ought to know where you do your backup (to your computer or to iCloud) and if you've done it both places then install whichever one is more recent.
You only have one iCloud backup, unless you specifically set your device up as a new device and backed up to iCloud (again) under a different device name.
The choice for most situations is between an iCloud backup and a local backup on your computer. How did you have your prior iPhone set up for backup -- to iCloud or to iTunes, and which did you do most recently if you've done both?