Learn Russian: Сегодня каждый бегун проходит мимо старого театра. - Today every runner passes by the old theater.

The prefix про- on a verb of motion generally means moving "through" or "past" something. Together with the preposition мимо (past/by), проходит specifically means the runner is moving past the theater.

Since ходить usually means walking aimlessly or making round trips, why is it used here for runners just passing by?

When you add a prefix to a multidirectional verb of motion like ходить, it simply becomes a regular imperfective verb. проходит is imperfective and is used here because the action is in the present tense and happens repeatedly ("every" runner). To say a runner will pass by once in the future, you would use the perfective counterpart пройдёт (which is built on идти).

Why do we say старого театра instead of старый театр?

The phrase is in the genitive case because it follows the preposition мимо (past/by). In Russian, мимо always forces the following noun and its adjectives into the genitive case.

Could we drop мимо and just say проходит старый театр?