For any stairway race, it's important to train in a real stairway, as opposed to just hopping on the stair climber at the gym. You can mix up your training with an occasional machine-based workout, but seek out stairs for the majority of your training so you're prepared and feel comfortable on race day.
Two or three times a week, perform intervals in which you run up several flights of stairs, walk down, and repeat. Each trip up and down is a repetition. Depending on your fitness level, start with a minute ascent, repeated for 4-6 repetitions. Add 30 seconds to your ascents every week to keep improving, and build up your total time based on the length of your race.
Running stairs will help you build leg strength, but you also need to work on endurance for a long race. Go for a longer run at an incline of 5-8%, or ride hills a couple times a week. You'll boost endurance for the race while giving your mind and body a break from the stairs.
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