What if the door was lying on the ground, not having been fixed to the wall yet, and you would have to literally step over it? Would it translate into Hungarian the same way?
"Átlép" is not correct in case of a door. It means that you step over the door (the door is lying on the floor).
The correct translation is: A fiatal turista belép/kilép az ajtón. (steps in, steps out), or "átmegy" (passes through)
The poor young tourist does not know yet, how Hungarians come in: jumping first into the garden, flying onto the chimney and crawling down the chimney. :-)
Why is it "on the door" instead of just "the door"?
When you use át-verbs to express going across something, the thing that's being crossed usually takes the -n ending.