When Makyura is sent to the graveyard, can you play a trap like Michizure from your hand, or would you miss the timing because his effect activates when he's already in the grave? I have a feeling I just answered my own question, but can someone give me a definite answer?

Makyura is a Continuous Effect, so it does not "activate", so you do not miss the timing for anything.

So assuming that nothing else occurs immediately after Makyura hitting the Graveyard, assuming it was sent from the field as Michizure requires, then there's no reason why you couldn't activate it unless some other effect is preventing you from doing so (Cold Wave or something)

Ah, ok, that would explain it. For some reason, I was thinking that it was a Trigger effect, or something...

Probably because of Jetroid or the games...anyways...very first rule on UDE

"This card's effect is a Continuous Effect."