Hi,
I know this is a couple months late, but just in case anyone else needs this info I thought I'd share it here. I updated my iphone 5s to iOS 9, but am still running snow leopard on my mac (which means I can't download the newest itunes and can't restore from my computer). I was able to successfully find my last backup file (in Library->Application Support->Mobile Sync) and copy the folder onto an external hard drive. I then transferred the entire folder from my external hard drive onto my friend's laptop (who has the newest version of itunes) and pasted it into their Mobile Sync folder.
Now this part is key, before you connect your device go to Itunes->Preferences->Devices and click the box next to "Prevent iPods, iPhones, and iPads from syncing automatically" This let iTunes find your device, but not automatically start a sync with a different apple ID. Go to the summary page and click on restore from backup (in the middle section on the right), the back up that you manually added should appear (at least it did for me!). Then when it says your device is restarting, it will appear again in itunes in a minute, disconnect it from the computer and follow the directions to update on the device. (it asked me if i wanted to allow some of the new features etc.)
Another important note, it didn't download all of the apps that I had on the phone before (maybe because my friend doesn't use them, I'm not sure) but I was able to restore all my data by re-downloading them manually on the iphone from the app store (go to the Updates tab, click on purchased, then choose "not on this device"). After I re-downloaded them, all of my stored date was still there (chats in whatsapp, saved settings and preferences, etc.)
I am not at all an expert, but it took me literally 8 hrs and a lot of frustration to figure this out, so I hope this post might help spare someone else from going through the same!