if I reveal a hidden "back off" and my opponent let it resolves, can he play an action which is intended to be used during the showdown

Yes — as long as the chain from Back Off actually resolves during a showdown, the state returns to Showdown Open and the next player in turn order gains Focus and Priority, so your opponent can play an Action spell.

Back Off played from Hidden has Reaction, but that only governs when it can be played — resolving it doesn't close or end the showdown. Per Rule 347.1.b, once the chain closes, Focus passes, and the player with Focus may play any legally timed card, including an Action. The showdown only ends if all players pass Focus in sequence (Rule 347.2.a). Just remember that the Action can't be played while Back Off is still on the chain, only after it resolves.

Caveat: if you played Back Off from Hidden outside of a showdown (for example, in reaction during your opponent's turn or in a Neutral Closed state), there's no showdown window afterward — your opponent only gets Reactions while that chain exists, and no Action window once it resolves (FAQ #1638).