Will I have to clear customs on my crossover flight in Sydney?

Customs relates to goods (and money), not people, and mostly for imports. So other than a few random spot checks, inspection will most likely only happen in Auckland, not in either Brisbane or Sydney.

What will happen in Australia is emigration formalities (passport control). This would most likely happen in Sydney, though there may be special cases I’m not aware of (you would need to tell us the airline and flight numbers).

With both flights on a single ticket/booking (though I have failed to find anyone selling such an itinerary), one hour to deplane, change terminals, go through passport control and security, and board seems quite tight to me, but if the airline sold the connection then they think it’s doable and they will usually be responsible for taking care of the passenger if the connection is missed (including rebooking on the next available flight, hotels if needed, etc.). But that means one may be delayed quite a bit, especially if flights are full (which used to be a big issue during holidays, not quite sure how that works out nowadays... less passengers but also less flights means less flexibility). With both QF and NZ not even operating one flight a day on that route, it’s still a cumbersome proposition.

I believe some airlines have specific procedures in place for domestic/international connections in Sydney to streamline the process, you should check their website for details.

If the two flights are booked separately, then it’s completely impossible if the passenger has checked luggage: one would need to deplane, wait for luggage, transfer to the international terminal, check in/drop bags before the check-in deadline, go through passport control, security, and board. The only flights I’ve found from SYD to AKL these days are flown by Qantas and Air New Zealand, and both their check-in deadlines are 90 minutes for international flights. So, not possible, at all. Don’t know if there are any other airlines operating such flights and what their check-in deadlines are, but I doubt they could be in any way compatible. And remember, that’s with the first flight operating on time.

With flights booked separately but no luggage, this will depend on whether online/mobile check-in is available, but remember that if the connection is missed, it’s the passenger’s sole responsibility to rebook (including paying change fees if the ticket is flexible or buying a brand new ticket if not — remember that a last-minute one-way ticket can be much, much more expensive than a round trip booked in advance) and pay any incidentals (e.g. hotel if one has to stay overnight). Also if there are more flights on the ticket (e.g. a return flight), those may be automatically cancelled by the airline (you usually have to fly all segments in sequence). Personally, I would definitely not do it, especially these days.

There are direct flights from BNE to AKL (not every day, but several times a week). That seems like a much, much better idea to me these days.