Is anything keeping you from just reinstalling the system and mounting your home into it again (maybe the majority of your customisations live in /home too)? I feel that is a lot less of a hassle than copying files around.
In principle you should be able to restore your system by just copying all of the relevant files from the backup to their correct partitions - it can’t really get any worse if it doesn’t work.
For the future: A backup is only any good if you know how to restore it and tested that that actually works.
Regarding the permissions: If you do a cp fileA.txt fileB.txtfileB.txt will normally be owned by the creating user. So a sudo cp … will create the files as root.
I would personally use rsync with a few additional options, archive among them. This way the fs is restored exactly as it was. But that doesn’t make a whole lot of sense if the files weren’t copied that way too.