Is "Kilenc fontos híd Budapesten van" incorrect? I assume that the different word order has a different emphasis, but are they both correct? Or does the English sentence imply a certain word order for the Hungarian sentence?
I would say yes and yes and yes.
Your version sounds like "Nine of the important bridges are in Budapest." Maybe they went there for a conference of bridges. Or we are taking an inventory of the important bridges of the world.
Is that because "Budapest" comes right before "van" in my translation? Whereas if "van" comes right after "híd" then you read it as "In Budapest ... 9 bridges are... "
Kind of, yes. Try to always see it this way, always ask yourself this question: what is in front of the verb? Because the position in front of the verb is the emphasized position.