I am arriving to New Zealand on 24th of December. Are there any issues with public transportation during the Christmas period that I should consider? I am especially interested in the bus timetables (are they same compared to non-holiday days)?
New Zealand doesn't treat Christmas eve as a holiday, this year the 24th falls on a Monday so it should be service as normal. Unlike some countries, New Zealand in my experience does offer some form of public transport everyday of the year in the main 3 centers. Id suspect Dunedin and Hamilton would also.
Auckland I have only spent a short amount of time in Auckland and their public transport was a painful mess at the time. Apparently its better, I couldn't find a simple policy on the maxx site. I did find some info about train disruption for a month starting December 20th which might be useful. I have checked a couple of bus time tables for Auckland and at the bottom they have a Sunday / Public Holiday detailing the times for the route. but...
I can only find media releases for Christmas 2010 services.
Christchurch: metro info has a very clear blanket policy for all services on all public holidays.
Wellington: the metlink site has a clear overview of Christmas services which start on the 25th, so again you will be fine at the very least on the 24th.