Unfortunately, на can often mean both, so I guess we have to rely on context for this one...this is a tricky one because being "at" the bridge is just as likely as "on" it. I think "at" should be accepted
The group of nouns that take "У" instead of "E" are usually monosyllable, like "на мосту", "в лесу", "на полу", with some exceptions such as "на берегу". With enough practice it just becomes second nature to say "на полу" and whatnot.
на мосту actually is prepositional case for the noun мост. There are a number of nouns that take the у ending in prepositional case instead of e, see a few of them given above. You just have to learn which takes у and which takes е.