Can you get fit by just walking the stairs?
Walking or running up and down stairs will certainly improve your cardiovascular fitness and add strength and endurance to your legs.
Depending on the size of the stairs you may be able to take them two or three at a time. If you wanted to and were in a suitable place, you could carry a set of dumbbells to intensify the workout. Like when sprinting uphill, even without dumbbells, the exercise will also act as a shoulder workout when you drive with your arms to aid the momentum.
I have just read the book of the great footballer Kevin Keegan, who followed a teammate's example when playing for Scunthorpe of running stadium steps carrying dumbbells. This was after noticing that the man had giant calf muscles.
Compared to somebody who doesn't exercise and even a person who does formal exercise at a gym and doesn't push themselves, an individual who regularly uses stairs is going to be much fitter, often with the bonus, when doing so at work, of being paid to exercise!