You need to first update your computer to 10.6.8, then you can update iTunes to 10.7 & sync your phone. The iPhone 5 requirements are printed right on the box:
Syncing with iTunes on a Mac or PC requires:
- Mac: OS X v10.6.8 or later
- PC: Windows 7; Windows Vista; or Windows XP Home or Professional with Service Pack 3 or later
- iTunes 10.7 or later (free download from www.apple.com/itunes/download)
You can "try" this:
iTunes 10.6.3 was released for developers to use with iOS 6.0. I cannot guarantee this will work for you, but other than updating to SL, it is your only other choice.