Coincidence I found this thread. Just the other day I ran up six flights of stairs at the metro, and felt fatigue only on part of the last flight. I reflected later on how tired I used to get doing this.
I attribute my strength to my leg exercise routine, specifically regular squats, lunges and jump-squats. You really should research the net on finding good pictures for these exercises as improper form could really hurt your knees. Start slow, and work your way to 3 sets of 15.