Can you cast instants during your opponent's upkeep before their upkeep triggers resolve?

When the upkeep step begins, any 'at the beginning of upkeep' triggered abilities trigger and are placed on the stack. After all triggers are put on the stack, the active player (your opponent) receives priority first. Once they pass priority, you receive priority and may cast instants or activate abilities before anything resolves (CR 117.3b, CR 503.1).

However, there is a narrow window before triggers are placed on the stack where no player has priority — the game simply checks for triggered abilities. You cannot act in that instant. Your first real opportunity comes once all upkeep triggers are on the stack and priority is passed to you (CR 117.5).

Once you do have priority, you may cast any instant-speed spell or activate any ability with an instant-speed activation. Your opponent's upkeep triggers sit on the stack waiting. If you cast something, it resolves (assuming no responses), and then those triggers still wait to resolve after (CR 405.5).

Concrete example: Your opponent controls Azorius Arrester with no relevant triggers, and you have a Counterspell in hand. On their upkeep, suppose they have a triggered ability from Howling Mine that triggers. After it goes on the stack and your opponent passes priority, you can cast Counterspell targeting it — it resolves first, countering the draw trigger before it ever resolves.

In short: you cannot act before triggers are placed on the stack, but you absolutely can act after they are on the stack and before they resolve, as long as you receive priority (CR 116.3, CR 503.1).