MtG question about Mimic Vat?

It depends on whose turn it is. Since you cast Day of Judgment, a sorcery, I assume it's your turn. When multiple triggered ability trigger at once, the active player puts theirs on the stack in any order, then the non-active player does the same. That means the non-active player's ability will resolve first. So in this case, the Messenger would undie, coming back with a counter under your opponent's control before you can imprint it on the Vat.

If it were your opponent's turn, the opposite would happen.

When triggers controlled by 2 different players are put on the stack at the same time, they go on the stack in APNAP order (active player/non-active player). Since you are the active player (the player whose turn it is), your trigger goes on the stack first and then the GM trigger is put on top of it. Last on the stack comes off first so Geralf's Messenger would return to the field before Mimic Vat would be able to exile him.

Long Story Short, you do not get Geralf's Messenger onto your Mimic Vat. The way to make it work would be to use Go for the Throat (or some other such instant speed removal) on it on your opponent's turn instead.

Exiling the card as the first ability resolves is optional. If you choose not to exile it, or you can't exile it because the card has somehow left the graveyard before the ability resolves, the ability simply doesn't do anything as it resolves. Any card currently exiled by Mimic Vat remains exiled.

So the trick would be if your ability resolved before Geralf's Messenger's.

So we go to triggers. AP/NAP I think is the order. Since the stack goes Last-on-first out, well, that's bad for you unless you have a stifle.

Then you could counter the undying trigger.