Must have been a headache!

Thanks for clarifying your trip details. So on 7 April Emirates announced a reduction for their EK171 service to Kiev effective 2 May and said affected bookings will be accommodated onto the remaining 4 weekly services. ( http://goo.gl/16FaMz )

I'm assuming Emirates contacted you to advise you of the changes. In such situation Emirates offer to either: waive cancellation fees providing full refund, or adjust your reservation accordingly. What needed to be done there is to request to depart Auckland the previous or the following day. If Emirates refused then a full refund would be in order.

Now regarding the flight times to/from Kiev, these were changed by 35 minutes for going out and 2hr 45 min on the inbound (departs after midnight instead of just midnight), at least according to the daily schedule released back in July 2013.

I think the major issue you have here is how to get your travel dates adjusted as a result of the service cancellation. The responses provided by the AKL staff do not resonate well.

You need to put this into writing, specific simple point email to their AKL office. Something like: In reference to booking XYZ on 27 May, the new travel dates do not suit I would like a full refund.
Or if the major issue is getting there on the 28th, then either ask to be accommodated onto the WAW flight on the 28th (free of charge, and the flight is wide open) OR have you depart on the 26th.

Now even on the 26th/28th departures from AKL there will be 12 hours in Dubai so hotel accommodation needs to be provided (or last time I checked at least).

Others will chime in with their advice.