Hello and welcome to the Support Community
Yes would be the answer to your question. In order to have the latest version of iTunes for your iPhone to sync, you would need to upgrade your OS from Mac OS X 10.5.8 to Mac OS X 10.6.8
I'm lead to believe that copies of Mac OS X 10.6 are available from Apple at about $20
Alternatively a copy on DVD can be obtained from online vendors such as Amazon at a higher cost.