Can I react with hidden Edge of Night to equip a unit (both on same battlefield) that is being pulled by Irresistible Faefolk?

Yes — play Edge of Night from facedown as a Reaction to Faefolk's move trigger, and it attaches to your unit before the pull resolves, so the unit is dragged over with the gear still on it (attached cards move with their top-most card, Rule 718.5.c).

The attach isn't the [C] Equip ability — that's an activated ability, so Main Phase/Open State only and unusable here. It's Edge of Night's own play effect ("When you play this from face down, attach it to a unit you control (here)"), which resolves as the gear resolves, on top of the Faefolk trigger (FAQ #12267). Two requirements: it must have been hidden on a previous turn (it only gains Reaction the turn after — Rule 811.6), and the unit must still be at that battlefield when Edge of Night resolves, since the play effect can only pick a unit "here" (Rule 811.1.d.3).

Two cautions. If your opponent responds and the unit leaves that battlefield before Edge of Night resolves, the attach is ignored and Edge of Night sits unattached there until it's recalled in the next Cleanup. And this doesn't save the unit from the pull — Faefolk's target locked in at finalization, so it still moves, just at +2 [M].