Yes, you have to upgrade to OS X 10.6.8 Snow Leopard to install the current iTunes and sync the iPhone 5.
As long as you have at least 1GB of memory and some free disk space, call the online Apple store, which in the US is at 800-MY-APPLE, or it can be ordered online from Apple too. $19.95. Only available on DVD. Install, run Software Update until there are no more updates, then install the current iTunes.