Terminal 2 is the Star Alliance Terminal in MUC, mostly LH of course. They have two levels, gates H (on top) and gates G (below). Arriving from overseas you will probably arrive at an H gate and depart for FCO from G. I have never been to the FCL but I think you can use it from both levels.
To FCO it will probably be an aircraft of the A320 family. If it is parked at an outside parking position you will get the sedan service (S-Class Mercedes or Porsche Cayenne; if they still use those next year).
If the flight departs from a gate position there is no sense driving there. Can just walk a few meters and board.
Hope you have a good flight and a good time in FCO!
Edit: Map of MUC T2:
http://www.lufthansa.com/cdautils/me...dia_697064.pdf