After you exit the airport security area in Terminal 1, turn left (right side leads you to the luggage claim area) and walk all the way to the South end of Terminal 1 (passing United, Southwest Airlines check-in counters).  Then exit through glass doors to an area where two automated UTA Trax (light rail) ticket boot stands are. (see map location marked as 1).

Purchase the ticket (one Trax fare per ride for seniors is $1.25) and take the Green line train to Courthouse (see map location marked as 2).

Get off the Courthouse and wait for the Red line train across the platform to the Fort Douglas stop (see map location marked as 3) . Relevant red line stops are Library,Trolley. Stadium, which are the two stops before Fort Douglas, andUniversity Medical Center is the last stop.

Get off at the Fort Douglas stop. On the right side of the platform,  you will see a sky bridge.  Walk toward the elevator below the bridge and take the elevator upstairs connected to the head of the bridge. Walk across the bridge to the other side and then follow the sign to the University Guesthouse,  walking along sidewalks (bear left).  The total time from the trax platform to the Guesthouse should be around 5 minutes.