According to Apple's official list here http://www.apple.com/osx/specs/ your Mac mini is compatible with Mountain Lion which of course means it is also compatible with Lion.
The upgrade process is to first upgrade to Snow Leopard, this gives you the ability to use the App Store. Then you should be able to buy and download the Mountain Lion installer (the Lion one is no longer available to purchase).
You should call the Apple online store at 1-800-MY-APPLE and ask for Snow Leopard, which may be on their legacy products list. It's never been available via download. Otherwise, look at Amazon or hardcoremac.com.