Does "apropos" take a preposition? How do you use this word, anyway?

The answer to your either-or question is "yes". :)

In other words, apropos can be an adverb, adjective, or preposition on its own, or it can be used as part of the prepositional phrase apropos of.

Depending on how you want to use apropos, synonyms can include "by the way", "incidentally", "regarding", "concerning", "opportunely", or "appropriately".

The 'a' in 'apropos' already includes the 'of'. The original French is "à propos".

To me it sounds very strange to use "Apropos of Easter, have you bought eggs yet?", since I use "apropos" like "Speaking of Easter, have you bought eggs yet?", so the "of" is not required.