Here is my two cents. If by

He ran down/up the road (to an unspecified location not far from here)

you mean

He ran along the road

then

Il a couru le long de la route.

Il a couru sur la route pas loin d’ici.

won’t convey the meaning you are looking for because it can just mean that he ran on the “nearby road”.

As you have pointed out the difficulty here is to convey the meaning of “up”¹. We can do this in different ways depending on the surroundings, in all cases we have to use a periphrase.

• Il a longé la route/rue sur quelques mètres en courant.
• Il a parcouru quelques mètres en courant le long de la route.

In this last sentence if the context is clear and we already know he’s on the road, you can leave out le long de la route. (Il a parcouru quelques mètres en courant.)

If the road is bordered with houses you can say:

• Il a couru à quelques maisons d’ici/jusqu’aux boutiques...

in which case you could use courir as a conjugated verb. But as often when trying to translate verbs of movements followed by a preposition into French is to use a gerund to convey the movement.

_{¹ Although “he ran up the road” could also mean he ran “upwards”.}

I think the meaning is best conveyed by using remonter and descendre. It doesn’t imply height elevation.

Il a remonté la rue en courant.
Il a descendu la rue en courant.

Similarly “she ran up/down the stairs” would be Elle a monté/descendu les escaliers en courant. However that does imply vertical movement.

Pourquoi “remonter” plutôt que “monter” dans le cas de la rue ? Je ne sais pas mais c’est ce qui me semble le plus naturel. Peut-être que “monter” ne s’utilise que quand il y a une différence de niveau ?