Mountain Lion on MacMini 1,1 Possble?
Not without modifying the installer and dealing with any issues / drivers / recompiling some of the code as needed.

The list of Macs that Apple checks before installing is on the requirements page:


http://www.apple.com/osx/specs/


As that link could easily change within a year's time, here for posterity are the models listed as supported on Mountain Lion:


iMac (Mid 2007 or newer)
MacBook (Late 2008 Aluminum, or Early 2009 or newer)
MacBook Pro (Mid/Late 2007 or newer)
Xserve (Early 2009)
MacBook Air (Late 2008 or newer)
Mac mini (Early 2009 or newer)
Mac Pro (Early 2008 or newer)