Anyway, is this right?
Unfortunately, no. The particle you need is を, not に. 下に would mean downwards.
Making sense of transitive usage of 行く and 来る - 「を行く」 and 「を来る」
この道をまっすぐ行ってください。 Why を and not で?
下を走った橋 should be understandable if there's enough context, but if you worry about ambiguity, it may be better to explicitly add the subject of 走る and say ...