The following guide demonstrates how to transfer iPhone apps along with app preferences and documents from one iOS device to any other iPhone, iPad or iPod Touch. The process involves the use of CopyTrans Apps or CopyTrans Shelbee.
Then connect the iPhone you want to restore the apps to and perform Custom restore. Here are detailed instructions on how to restore the apps with CopyTrans Shelbee.
It's worth noting that if you want access to your old preferences (Wi-Fi logins, Settings and so on), third-party app data (game saves), health/exercise data or text messages, you'll have to erase your new iPhone and set it up again, this time selecting a backup of your previous iPhone to restore from. More transfer advice can be found in How to transfer contacts from iPhone to iPhone and How to back up and transfer text messages to a new iPhone.