Can you cast an instant during an opponent's end step after they pass priority?

During any step or phase, after a player takes an action (or passes priority), priority is passed to the next player in turn order. If the active player passes priority during their end step without taking an action, each other player in turn order gets the chance to act before the game moves on (CR 116.3).

When priority reaches you, you are free to cast any instant-speed spell — including instants — or activate any ability that can be activated at instant speed (CR 116.1, 307.1). The end step is not special in this regard; the same rules that allow you to cast instants during combat also apply here.

One important nuance: after you cast your spell, priority returns to the active player first, then passes around the table again. The end step does not end until all players pass priority consecutively without taking any action (CR 116.4).

Concrete example: It is your opponent's end step. They tap out to activate an ability and then pass priority. You cast Counterspell targeting that ability's spell. This is perfectly legal — you received priority at instant speed during the end step and used it.

Note: if you cast a spell during the end step that causes a triggered ability to trigger, that trigger will be put on the stack and resolved before the game moves to the cleanup step, keeping the end step going (CR 513.1).