Downgrading a new mac mini 2012 from Maverick to Mountain Lion
According to the support article listing minimum versions of OS X per computer, your Mac can run OS X 10.8.1


http://support.apple.com/kb/ht1159#Mac_mini


Since you have AppleCare, you can actually call them and ask how to install a specific version of the software and be supported on how to install things.

In a nutshell, you'll have to use Internet Recovery to wipe the drive and get back to the version of OS that shipped with your Mac. From that point, you can download the older OS from the App Store (AppleCare may have to issue you a redemption code for that build if you don't have it for your Apple ID).

From there, you can partition your internal drive and install Mountain Lion on a separate partition or install it onto a USB drive.

At that point, you can restore your data and settings on to Mountain Lion and/or wipe the internal drive booting off the external to finally install Mountain Lion as the only OS on the internal drive.

It's a bit of an exercise, but you can ask here or from Apple for guidance on each step along the way if you get lost.