You should upgrade to Mavericks, it's also a free. The technical specifications for the upgrades, to Mountain Lion and Mavericks are identical.
You're confusing 'updates' with 'upgrades. It's displayed in the panel on the right. To find the model identifier open System Profiler in the Utilities folder. It would be preferable to use Ethernet because it is nearly four times faster than wireless. The file is quite large, over 5 GBs, so allow some time to download. To upgrade to Mavericks you must have Snow Leopard Download Mavericks from the App Store. You can upgrade to Mavericks from Lion or directly from Snow Leopard.
