Can I tap Irelia legend when I select her during a showdown with a spell?

Yes. Playing a spell that chooses Irelia, Fervent during a Showdown triggers Blade Dancer's ability ("When you choose a friendly unit, you may exhaust me and pay [A] to ready it"), so you can tap the Legend and pay a Rune to ready her (FAQ #9382).

Note the extra payoff: readying Irelia, Fervent also fires her own "when you ready me" trigger, stacking another +1 Might on top of the +1 she gets from being chosen. If she's already readied, the ready does nothing (Rule 415.1.c), but you can still pay.