How long does Vex Apathetic stun last? If I stun them during their turn, when it's then my turn (after their turn), is it unstunned?

Yes, it's unstunned by the time your turn starts.

Here's the timing breakdown:

1. Your opponent plays a unit on their turn → Vex triggers and stuns it.
2. The next Ending Step is your opponent's own Ending Step (same turn). Per Rule 423.1.a.2, Stunned units lose the status at the beginning of that Ending Step. As FAQ #3710 confirms, this means the unit becomes unstunned during the same turn it was stunned.
3. By the time your opponent's turn ends and your turn begins, the stun has already expired.

Important distinction: The "can't move it this turn" clause (from Vex's text) is separate from the Stunned status. That restriction runs until the Expiration Step of the opponent's turn (Rule 317.2.c), so it also expires before your turn starts. Your opponent simply cannot move that unit for the rest of their turn, but by your turn, neither the stun nor the movement restriction is still in effect.