If I play Gutter Guardian from face down, can I show off itself?

No — Gutter Guardian can't show off itself when you play it from facedown.

Show Off is defined as revealing the object from your hand or picking it from among friendly game objects on the board (Rule 832.1.c). While Gutter Guardian is being finalized it's a chain item — not a card in your hand and not yet a game object on the board — so there's nothing to pick. The card's own reminder text reinforces this with "pick another friendly one on the board."

So to turn on its "[Add] [A]" rider you need a different qualifying object: another card with Hidden in your hand to reveal, or a face-up friendly card with Hidden already on the board. Note that a facedown hidden card does not qualify as "a card with Hidden" (Rule 811.5.a), so your existing facedown cards can't be shown off either.