You have options. If you can just restore by sync with iTunes that would probably be the best option. Sometimes restoring the backup puts the problem back. If you sync you should get your apps back and hopefully contacts (whatever you have setup to sync basically)
However, you can also restore your backup and try it. If that works great, if the problem returns THEN restore again and do the sync option. Its you to you. some folks restore their backup and have no issue some the issue returns. Keep in mind also that it could be some app you have causing the problem. Since the device is working in its out of box condition. You may want to consider installing an app or two at a time to see.
Bottom line, it appears your device is ok so determining what was causing the issue is the next step. Either corrupt iOS update which you resolved by restoring or some app that is misbehaving.
If it were me, I would try and restore the iCloud backup. Nothing to loose but some time. If the issue comes back you can always wipe it out again go with the other option using sync.