For Vex Apathetic ability, does it stun the unit only for the turn that it is played or for how long?

The stun applied by Vex, Apathetic lasts until the beginning of the next Ending Step.

According to Rule 423.1.a.2, stunned units lose the Stunned status at the beginning of the next Ending Step. As clarified in RiftJudge FAQ #3710, this refers to the next Ending Step that occurs in the game, which is the active player's (the player who applied the stun) Ending Step. This means the unit becomes unstunned during the same turn it was stunned.

Additionally, Vex's ability text explicitly states: "They can't move it this turn," which reinforces that the restriction on the unit's movement is limited to the turn in which it was played and stunned.