Fairly easy. At first you'll just have to attack mandarin as you would the minor bosses, nothing special. Eventually he will teleport him self into a room blocked off by a shield. Now there are four teleporters in the room, one for each wall. You can't simply teleport to Mandarin, you need to draw the little bug-robots he sends after you into these teleporters(the one you need will be shining). The bugs will be teleported and explode, killing the Ultimo in the room. With the Ultimo gone, you are free to teleport to Mandarin. Proceed to attack him until he teleports into one of the other rooms. Repeat the process until Mandarin teleports back downstairs. Now all you need to do is finish him off.