Yes, because a phase or step does not end until all players have passed priority while the stack is empty.
So, after your opponent's Enlightened Tutor
(for examle) has resolved, the active player will receive priority. If they don't have anything else to do, they'll pass priority and it will go to the next player, and so on. You'll get priority to use Archive Trap
eventually, and you can do so for its alternative cost.
Also, in the future, please start a new thread for your questions, as it's generally frowned upon to revive threads that are over a year old.