I think cycling will help your cardio on stairs - it just may take more than the 6 days or so of riding that you have done.
You may still get winded by running up stairs - especially if your job is to light a torch on top of the statue of liberty.
If you are really running up stairs, you are probably going anaerobic. I find that my commuting doesn't do much for anaerobic fitness unless I intentionally start sprinting up hills or mixing up your speed to create workouts. I've been commuting 75miles/week for 6 months, and have noticed great gains in aerobic fitness, resting heartrate, and blood pressure, etc. But... I'm hitting a wall w/ speed and anaerobic improvements, so I'm working on a routine to mix it up a bit.