The first time you setup your new phone, sign in using the same Microsoft account. You should be given the option to restore your phone. Now just tap the backuip you want followed by Next.
I think that I signed in using the same Microsoft account, but I didn't get the option to restore?
Do you know in what phase that option is possible?
Correct me if I am wrong but as far as I know you will need to hard reset to see that option again. However, you could already have most of your content synced which means you should be able to sync it your other phone.
Example, I use a 920 and an 820 concurrently. Both phones are always synched with each other by synching both with my Microsoft account and my Exchange server.
I managed to do restore with these steps.
1. New backup with my old 920.
2. Reseting the new phone
3. Booting the new
4. Login to Microsoft account
5. Selecting today's backup (there was listing of backups).
And that's it.