It is correct to say Je suis en ville. But incorrect to say Le bureau de poste est en centre-ville? per this lesson.

This is to do with the preposition of place used with feminine or masculine (place/location) nouns. Ville is a feminine noun - so “en ville”. Centre-ville is a masculine (compound) noun - so ‘au centre-ville’.

Hi Maarten, are you sure that's the reason? As I understand it, "en ville" means a general "in town".  You could also say "à la ville de ma naissance", in which case you would use the preposition "à". For example: "J'habite en ville" vs. "je vais à la ville de ma naissance." It is "au centre-ville" because this is a specific place.