1. Go to Settings > iTunes & App Stores and sign out of your Apple account.
2. Click on this link or if you're not viewing this on mobile, open Safari on your iPhone and search for "Pokemon Go app store" and click the link that will take you to the Pokémon GO app in the App Store. You should see a pop up saying that this app is not available in Singapore, but is available in New Zealand. Click Change Store to go the New Zealand App Store.
3. You may have to search for Pokémon GO again. Do so and you should see it. Click Get and it will prompt you to Sign In with an existing Apple ID or to Create a new Apple ID.
4. Create a new Apple ID and choose New Zealand as your region. Do note that you'll need to be able to verify your Apple ID to download the game, so make sure you have access to the email address you're using to set up this temporary account. Leave the billing information blank. When you get to the address details, use this website to generate a fake New Zealand address.
5. Verify your account details and log in to download the game. You can now sign out of your New Zealand account and go back to your regular Apple ID.