Train is an option, not too difficult (Stuttgart airport -> Stuttgart main station -> Munich main station -> Allianz Arena, should be no more steps), not sure about the financial part. Rental is an option too, but is most likely connected with some short-distance train or taxi in Munich from the nearest rental-dropoff to the Allianz Arena. If its a roundtrip, rental may be the better option in general. Car should be, but is not neccessarily faster, heavily depending on luck with traffic. Also parking at the AA is supposedly a bitch and will on a gameday take a while, and from what I've heard, will take ages to get out of again, making train overall the more reasonable option.
Let me look up the train connection real quick..
Assuming you want to get to a game that starts say 6pm, you would need to leave the airport by train shortly after noon, in order to arrive at the AA between 4 and 5pm, later trains get you there past 6pm.
Not sure if Bahn-links work properly when copied, but if so, this would be the connection and pricing for next saturday, local Munich-trains excluded (~5er ontop for a day-ticket):
http://reiseauskunft.bahn.de/bin/qu...arture&HWAI=JS!ajax=yes!&REQ0HafasScrollDir=2
So about 35-45 EUR one-way for a train ticket, ~4hrs time, 70-80 for the roundtrip. A one-day-rental with fuel is probably clocking in at the same plus fuel, should get you there within 2-2.5hrs though, except traffic on the way or on site strikes.