Does Vex Apathetic ability to stun target the unit? Does it stun a Rune Runner?

No, Vex, Apathetic does not target the unit it stuns, and it does stun a Ruin Runner.

Reasoning

• Programmatic Selection vs. Targeting: According to Rule 355.10.d, a game object is not a target if it is "programmatically selected based on its characteristics rather than chosen by the spell or ability’s controller." Vex's ability automatically identifies and selects the unit that triggered the condition ("When an opponent plays a unit"). Because the ability does not require you to make a choice or designate a target, it is not considered targeting.
• Bypassing Protection: Ruin Runner has the ability, "I can't be chosen by enemy spells and abilities." As established in RiftJudge FAQ #8772 and #8685, this protection only applies to effects that require a player to manually choose or target a unit. Since Vex's ability uses programmatic selection, it bypasses this protection and successfully stuns the Ruin Runner.

This interaction is consistent with established rulings for Vex, Apathetic regarding units with "cannot be chosen" protection.