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

Yes. Because of daylight saving time the gap moves between +11h and +13h.

When is the best time to call Dublin from Auckland?

There is no overlap between standard working days, so one side has to call early or late.

Auckland is 11h ahead of Dublin. This gap is not fixed: daylight saving time moves it between +11h and +13h during the year.

Auckland

New Zealand

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Dublin

Ireland

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A 9-to-5 working day in Auckland does not overlap one in Dublin at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Dublin · Ireland	Europe/Dublin	UTC+1

What is the time difference between Auckland and Dublin?: Auckland is 11h ahead of Dublin. This gap is not fixed: daylight saving time moves it between +11h and +13h during the year.
Does the time difference between Auckland and Dublin change during the year?: Yes. Because of daylight saving time the gap moves between +11h and +13h.
When is the best time to call Dublin from Auckland?: There is no overlap between standard working days, so one side has to call early or late.