iphone backup issues

Hello, I have an old macbook that will only support itunes 10. I finally upgraded my iphone 4s to 7.1.2 so now I can't sync it to my laptop. It needed to be restored during installation, so I lost everything. I did have it backed up on my laptop before I installed it, but since I can't sync it with my old laptop, how can I get my backup restored to my phone? Can I copy and paste the files to a windows 7 with the newest version of itunes and sync it that way? Or is there an easy way to restore my iphone to an earlier version without having to jailbreak it?

P.s. I can't install iExplorer (my computer is just outside the system requirements!

Thanks!!