Updating iPhone contacts with iPad backup

Sorry, but you can't extract single information from a backup.

However, syncing your contacts to iCloud should work.

Use Contacts on iCloud.com - Apple Support

In the settings for iCloud on your iPad enable iCloud Contacts, and open the contact app on the iPad and wait a couple of seconds to let the information sync to iCloud.

Then do the same on your iPhone and open the contact app. Your contacts should sync back to the phone.

You can disable iCloud contacts after that, if you don't want to keep on using iCloud contacts, you will be asked if you want to keep the info on your device. Tap on "yes".