I’m going to assume this whole thread is a joke troll. But anyway..
and i'm even a physics dude [...]
Technically speaking you're both doing the same amount of "work" since you're travelling the same distance.
What? No. Technically the dude is doing more “work”, proportional to their masses.
http://en.wikipedia.org/wiki/Work_(physics)work = force * distance = 9.8 m/s/s * mass * distance
(assuming gravity at the surface of the earth)
You may want to break back open that high school mechanics textbook, physics dude.

I would assume the little girl would be using more energy.
Likewise, the guy is “using more energy”. Calculating the precise amount of energy used is a bit tricky, since it depends on the efficiency of the muscles, circulatory system, etc. We'd need more information about these people and a complicated model of human energy use to calculate it out.
But if all you care about is who uses more energy, it is definitely the man.
* * *
As for who tires out faster, or who could climb more stairs before exhaustion, or who could climb the stairs faster, that’s something you’ll have to verify empirically. At the world-class professional athlete level, a 5'11" 190 lb man is going to have more strength and stamina than a 4'5" 100 lb woman. But I know some short thin women who are amazing athletes and I know some 5'11" 190lbs dudes who can barely walk.