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

At the start of your opponent's upkeep, the game first puts all 'at the beginning of upkeep' triggered abilities onto the stack (CR 603.3). However, no player receives priority until after these triggers are placed on the stack. Once they are on the stack, the active player (your opponent) receives priority first (CR 117.3b).

This means you must wait until your opponent passes priority before you can act. If your opponent passes priority without doing anything, you then receive priority and may cast instants or activate abilities — and at that point the upkeep triggers are still on the stack and have not yet resolved (CR 116.3c, 117.3b).

So technically you cannot act before the triggers are placed on the stack, but you absolutely can act before they resolve, since each player gets priority before any spell or ability resolves (CR 116.4).

Example: Your opponent has a Upkeep trigger from Howling Mine that reads 'at the beginning of each player's upkeep, draw a card.' During your opponent's upkeep, that trigger goes on the stack. Your opponent passes priority; you may now cast an instant like Counterspell to counter the trigger before it resolves.

In summary: you cannot squeeze in a spell before upkeep triggers are placed on the stack, but once they are there, you can respond before they resolve, as long as you receive priority (CR 116.1, 116.4).