I would definitely recommend just taking the train: the red line goes directly from the airport (its in the same building, just follow the signs from the gate,to the main terminal, towards "ground transportation", and then down to where it says "rta") to the campus....you could get either off at the University Circle stop (south campus) or the East 120'th stop (a little off the edge of north campus).
The ride is about 45 mins and only $2.00 (one way)
Look here for maps and schedules:
Timetables, Maps & Schedules | Greater Cleveland Regional Transit Authority (you will want the 'red line' designation '66)