I bought a managed IAP (e.g. the Coin Doubler) from the iOS App Store, but how do I get that again on my new device?

Make sure you are logged into the App Store with the same account you used when you originally purchased the item.  Then  go to the Settings / Support screen and click "Restore Purchases".  
 
Please note that only managed items like the Coin Doubler or Go Premium can be restored.  Consumables like coin or gem packs cannot be restored.