If you're trying to sync the device with your iTunes library, you can do it without upgrading the OS.
If you need the iCloud pane, you need to upgrade the computer to at least 10.7.2. Open the Mac App Store and try downloading Mavericks. If you get told it's incompatible, choose About this Mac from the Apple menu, check if the computer has at least a Xeon or Core 2 Duo(not Core Duo) CPU and 2GB of RAM, and if it does, click here and order a download code for Lion 10.7.
Back up your data before upgrading. Mac OS X 10.7 and newer don't support PowerPC programs such as Microsoft Office 2004 or earlier.
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