I attack with Vex, Apathetic and Showdown stats. They react to my Showdown with Rengar AMBUSH. Is Rengar Stun or not?

Yes, Rengar gets stunned.

When your opponent plays Rengar via Ambush to Vex's battlefield, Vex's ability triggers immediately — "When an opponent plays a unit while I'm at a battlefield, [Stun] it." Ambush doesn't bypass this; playing a unit is still playing a unit (FAQ #9735, #10932). Vex's stun goes on the chain and resolves, stunning Rengar before combat happens.

Per Rule 423.1.b, a stunned Rengar won't contribute his Might to damage during combat, so Vex will deal 4 damage to Rengar while taking 0 in return.