It is near a river, but it doesn’t seem to be a river port itself:

Barrowton, too, is somewhat of a curiosity—a gathering place built at the foot of the reputed barrow of the First King, who once ruled supreme over all the First Men, if the legends can be believed. Rising from the midst of a wide and empty plain, it has prospered thanks to the shrewd stewardship of the Dustins, loyal bannermen to the Starks, who have ruled the Barrowlands in their name since the fall of the last of the Barrow Kings.

…The moon was rising over the wooden walls of Barrowton when they stepped outside. Reek could hear the wind sweeping across the rolling plains beyond the town.

Certainly, there aren’t any mentions of quays, docks, ships, or even boats. Indeed, if it wasn’t for the maps, you wouldn’t know that it’s near the water at all.