How do I sync music from iTunes 10.6.3 on a 2009 MacBook OS X 10.5.8 to a iPhone 5S?

My wife has a new iPhone 5s, which she set up from an iCloud backup. She’d like to connect to iTunes on her 2009 MacBook (Mac OS X version 10.5.8) so she can download her music to the new phone, etc.

When she plugs it into iTunes she gets an error that says “The iPhone “[Wife’s iPhone]” cannot be used because it requires iTunes version 11.1 or later. Go to www.itunes.com to download the latest version of iTunes.”

Okay, cool. I check for updates in iTunes first just in case and it says “This version of iTunes (10.6.3) is the current version.” so then I go to itunes.com and download the latest version of iTunes – iTunes11.1.3.dmg. So, I open the dmg and click “Install iTunes,” but then it says “This update requires OS X version 10.6.8 or later.” Hmm…

So, I try to do a Software Update to get her from 10.5.8 to a more recent version and the Software Update says “Your software is up to date.” So, then I remember hearing that the new Mavericks version of the Mac OS is free, so I try to upgrade to that, but it’s saying I need to do that from the Mac App Store, but it says I can’t download Mac OS X Mavericks from the Mac App Store without OS X 10.6.6 or later, so it feels like we’re in a bit of a loop here.

She wants to use iTunes (Apple product) to get her music onto her iPhone 5s (Apple product) from her MacBook (Apple product). How can we do this?