We've said this a thousand times on this forum, but here it is for you:
OLD OS + NEW EQUIPMENT = PROBLEMS.
Luckily for you there is an easy solution: simply call Apple, explain that you're on 10.5.8 and just got an iPhone 5. Ask if you can order a copy of Snow Leopard on DVD. They will charge you $29 plus shipping probably.
Once you get it, use it to upgrade your MacBook Pro. Do a backup first of course, and check to make sure all your mission-critical software is as up-to-date as possible BEFORE doing the upgrade.
Okay, now that you're on Snow Leopard, run Software Update repeatedly and install everything it tells you to until there are no more software updates. You'll now be running 10.6.8 and have the Mac App Store and the latest iTunes.
At this point you can now proceed to sync and manage your iPhone 5. If you wish, you can also proceed to Mountain Lion (assuming your machine qualifies for it, check first!!) for another $20 and be bang up to date. Mountain Lion (10.8, no you don't need to install 10.7 first) is only available through the Mac App Store.
