How do I move a deleted App to my old iPhone?
Did it stop working on the new iPhone because of the iOS update? Then restoring the app from the backup is not a solution, as it might upgrade iOS as well (do you know for sure what OS it was when you backed up?). 

If you are using the same Apple ID on your iPhone 5, you still can download and install the old app. Not really good idea sticking to the discontinued app, but still...  iMazing to my knowledge is the only software that can do it (disclaimer: I work for iMazing). Legit App Store downloads, and this functionality is 100% free at the moment. It should work if it was removed from sale by the developer. If it was Apple who removed it, most probably we won’t be able to help. 

Anyway, it’s quite straightforward, but I’ll post the steps just in case: 


Install iMazing &amp; connect your iPhone with the needed app,   
Select your device on the menu, and click Manage apps,
Locate the app in the apps list &amp; click the Download button
Sign in to your Apple ID to finish the download 
Once the download is finished, exit the wizard, connect your old phone, select it in the sidebar and click Manage Apps again. 
Select the app in question, and click Install - that’s it.