How can I sync the data from my iPhone 6 to my iPhone 4s using iCloud?

My iPhone 6 recently broke and I bought a cheap iPhone 4s to use temporarily. I'm on the white hello screen to the set up and when I get to the restore screen and try to click iCloud backup, I can't select a backup because my iOS isn't up to date and I can't update it because the operating system I need isn't compatible with the iPhone 4s.

You can't do a backup for the reason you stated. 4s isn't compatible. But you should be able to still download certain apps, songs, and some other stuff you purchased from Apple. So, individually but not as a backup.

Start by getting your contacts out of iCloud. Using your PC sign in to iCloud.com and select Contacts. Click the gear icon in the lower left and choose Select All. Click it again and choose Export Vcard. Mail the resulting .vcf file to yourself and open it on your iPhone.