Can you reveal Zhonya's from hidden at a battlefield in reaction to combat resolving (both players pass) or do you need to reveal it and pass action before combat can happen?

No — you cannot wait until both players pass during the Showdown to reveal a hidden Zhonya's in reaction to combat damage.

Here's why: once both players pass Focus in sequence during a Combat Showdown, the Showdown Closes (Rule 348). You proceed directly to the Combat Damage Step — damage is assigned and dealt simultaneously — then immediately to Combat Cleanup. Nobody gets a reaction window between "both players pass" and "damage is dealt." Hidden cards that weren't revealed earlier stay hidden through the whole process (FAQ #8150).

If you want Zhonya's to save a unit from combat damage, you must reveal it before both players pass — during the Showdown chain while you still have priority. This does reveal it to your opponent, but that's the cost of the timing.

One exception: If a unit has Deathknell (or any triggered ability that fires when it takes lethal damage), that trigger creates a new Chain window after damage is assigned but during Cleanup. You can reveal hidden Zhonya's in response to that trigger to save the same or another unit (FAQ #10434, Rule 323.4). But that requires a Deathknell unit to create the window — you can't create the window just by wanting to save a unit.