Syncing Contacts to iOS on Mac OS X 10.6
According to http://www.apple.com/icloud/setup/mac.html iCloud requires 10.7.5 or later. 

Since iCloud encrypts data between the client and the server, you won't find third party options. I would get whatever software you need on 10.6 upgraded and then upgrade the OS or set up virtualization to run 10.6 from within a newer OS if iCloud data sync is needed. 

Perhaps you could find a different cloud sync between to that's likely more work and aisle than the above.