The Big 12 has only had an a tie-in with the Fiesta Bowl since 1996 (the Bowl Alliance and the Bowl Championship), and as such is the weakest of the conference tie-ins. Before that, Texas was in the SWC conference which had a tie-in with the cotton bowl.
The Cotton Bowl would have probably had one of the following teams (remember that conference tie-ins were much less strict in older college football): Notre Dame, LSU, Georgia, Oregon, or Miami.
For the few years they would have played in the Fiesta Bowl ('96 through '98) I'd expect Notre Dame would be the opponent.
Also, remember that Penn State is a recent addition to the big 10, so it would probably USC Ohio State in the Rose Bowl, and Texas Penn State in the Fiesta.