Am I asking Laszlo this question or is it asking if it's Laszlo who's running? Or both?
Probably you are asking Laszlo who is running, but I believe both interpretations are ok.
Why do we need the -on suffix on varos? I'm not seeing where it fits in the translation.
város, like all nouns, needs to be in a case. Here it cannot be nominative case (city is not the subject doing the running) nor accusative (the city is not the direct object of fut).
As to choosing -on over other cases, the át-/-on pair seems to translate very closely to the English preposition through (as in the city is between the starting and ending point)