Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Dumaguete City to New Zealand as you normally would.
Dumaguete City (DGT), located 3.0 km from the city center, is the airport you fly out of when you book flights from Dumaguete City to New Zealand.
This is defined as Dumaguete City Airport (DGT) and Auckland Intl Airport (AKL). Codes like Auckland Intl Airport are often used to just call airports uniquely. Christchurch Airport (CHC), Wellington Airport (WLG), Queenstown Intl Airport (ZQN), Dunedin Airport (DUD), Nelson Airport (NSN) are some additional IATA codes for airports related to your trip.