Does Back-Alley Bar target the unit to give it +1 and can I use Irelia legend ability after that?

No, Back-Alley Bar does not target the unit, and you cannot use Irelia's legend ability in response to its effect.

Here is the breakdown:

1. Does Back-Alley Bar target?

No. According to Rule 355.10.c, a game object mentioned in the text of a triggered ability is not a target if it is included only as part of a trigger condition.

Back-Alley Bar's ability reads: "When a unit moves from here, give it +1 might this turn." The unit is the object that satisfies the trigger condition ("moves from here"), not a target chosen by the ability. Because it does not target, it does not trigger Irelia's legend ability.

2. Can you use Irelia's legend ability?

Irelia's legend ability (Blade Dancer) triggers "When you choose a friendly unit." Since Back-Alley Bar does not "choose" or "target" a unit, it does not meet the trigger condition for Irelia's ability.

Summary of Interaction:

• Back-Alley Bar: When a unit moves from the battlefield, the +1 Might effect is applied automatically upon resolution of the trigger.
• Irelia, Blade Dancer: Because no unit was "chosen" (targeted) by the Bar's ability, Irelia's ability does not trigger, and you cannot use it in response to the Bar's effect.