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

Yes. Because of daylight saving time the gap moves between +9h and +11h.

When is the best time to call Cairo from Auckland?

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

Auckland is 9h ahead of Cairo. This gap is not fixed: daylight saving time moves it between +9h and +11h during the year.

Auckland

New Zealand

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Cairo

Egypt

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

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Cairo · Egypt	Africa/Cairo	UTC+3

What is the time difference between Auckland and Cairo?: Auckland is 9h ahead of Cairo. This gap is not fixed: daylight saving time moves it between +9h and +11h during the year.
Does the time difference between Auckland and Cairo change during the year?: Yes. Because of daylight saving time the gap moves between +9h and +11h.
When is the best time to call Cairo from Auckland?: There is no overlap between standard working days, so one side has to call early or late.