Does the time difference between Auckland and Tokyo change during the year?

Yes. Because of daylight saving time the gap moves between +3h and +4h.

When is the best time to call Tokyo from Auckland?

Between 12:00 and 17:00 in Auckland, which is 09:00 to 14:00 in Tokyo.

Auckland is 3h ahead of Tokyo. This gap is not fixed: daylight saving time moves it between +3h and +4h during the year.

Auckland

New Zealand

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Tokyo

Japan

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A 9-to-5 day in each city overlaps for 5h. That is 12:00–17:00 in Auckland, which is 09:00–14:00 in Tokyo.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Tokyo · Japan	Asia/Tokyo	UTC+9

What is the time difference between Auckland and Tokyo?: Auckland is 3h ahead of Tokyo. This gap is not fixed: daylight saving time moves it between +3h and +4h during the year.
Does the time difference between Auckland and Tokyo change during the year?: Yes. Because of daylight saving time the gap moves between +3h and +4h.
When is the best time to call Tokyo from Auckland?: Between 12:00 and 17:00 in Auckland, which is 09:00 to 14:00 in Tokyo.