Can you cast a spell during your opponent's end step after they pass priority?

During any phase or step, players receive priority in turn order. On your opponent's turn, they receive priority first in each step, including the end step. If they pass priority without doing anything (or after resolving their spell/ability), priority passes to you. At that point, you may cast instants, activate abilities, or take any other action allowed at instant speed (CR 116.3, 500.2).

The end step is a regular step where the active player (your opponent) gets priority first. Once they pass, you receive priority and can act. If you do nothing and also pass, the game moves to the cleanup step (CR 514.1). But if you cast a spell or activate an ability during the end step, the game stays in the end step until all players pass priority in succession with an empty stack (CR 116.4).

One important nuance: you can only cast spells with flash or instants at this time, since you are not the active player and it is not your main phase with an empty stack (CR 116.1a, 307.1). Sorcery-speed spells cannot be cast here.

Concrete example: Your opponent ends their turn and passes priority during their end step. You respond by casting Opt (an instant). This is perfectly legal. After Opt resolves, your opponent gets priority again; if both players pass, the end step ends and the cleanup step begins.

So the answer is a clear Yes — as long as what you want to cast or activate is legal at instant speed, you may do so after your opponent passes priority during their end step.