Can i use shuriken flip in an ongoing showdown? Since shuriken flip doesn’t have a action keyword means if my opponent play a card with action goes first? For example, i sent two units from a occupied battlefield then a showdown started, i used my empowered akali legend to pull one of my unit there sending it to my base ready, can i use shuriken flip to send that unit back to the showdown where he left and fight with them?

No. Shuriken Flip can't be played during the showdown — it has neither [Action] nor [Reaction] (only Flow, which just adds the play-from-trash option), and during a Showdown State only Action/Reaction cards are legally playable (Rule 343.1.a; FAQ #8041). Same for the unit at your base: it can't Standard Move back either, since moves are banned during a Showdown/Combat (Rule 144.1.c). So the ready unit stays at base until the showdown/combat is fully over, and it won't be re-entering that fight.

On ordering: yes, after Rogue Assassin's activation resolves and the chain empties back to Showdown Open, Focus passes to the next player in turn order (Rule 346) — so your opponent gets the first chance to play an Action spell before you act again. But that doesn't matter for Shuriken Flip: even when Focus returns to you mid-showdown, it still can't be played.

Once the showdown/combat ends and you're back in Neutral Open on your Main Phase, you could play Shuriken Flip to send that unit to the battlefield — but by then there's no fight left to join.