I'm certainly not the best travelled person on the forums, but I have been to a few of the NZ airports...
To answer your questions:
1) Is the destination as close to the airport as you imagined it would be?
Yes, but then I did do some research before travelling. For Auckland and Christchurch, we were picking up and dropping off a campervan, so didn't have far to go to get to the rental site (and shuttlebuses were laid on by the rental company). For Wellington, we were going to a campsite that we knew - we knew that it was a way out of the city (it's not actually in Wellington, but just outside an area called Lower Hutt)
2) Is transport laid on? Is it reasonably priced?
Yes, as far as I am aware, there are buses and taxis from all three of the airports that I've visited - we did debate getting the shuttle bus from Wellington, but as it didn't go to the campsite, meaning we would have to change in the centre of Lower Hutt, with our heavy luggage (the nice people at the airport allowed me to have slightly overweight baggage

), whilst jet-lagged (the flight, via Hong Kong, with a change in Auckland, was about 27 hours long), we decided to go by taxi! The taxi that we took was $55 (appx £20) for two of us - looking at the Wellington airport website, it's about $30 to get you to the centre of the city. For comparison, the regular Flyer Bus is $8 per person to get to the city. For other airports, its worth checking their websites.
For the rest of transport, it does depend on where you are and where you are wanting to go. For us, in Lower Hutt, we have excellent transport facilities - a revamped train service that runs every twenty minutes/half an hour (depending on time) and is about a 20 minute journey into Wellington Central. It is reasonably cheap, and there are mulit-travel discount cards, too. The bus network is good, though quite a few buses only run weekdays and in commuter time (i.e. trying to get places after 6pm gets a bit harder!) But, again, they are reasonably priced, and, aside from one journey when a couple of buses didn't show, they run on time

However, I know that trying to get between cities is a bit harder - the trains between Wellington and Auckland, for example, only run a couple of times a day, and cost the same as to fly (very green!

)
I've only been here since November, so I don't know what say the people around the airports had with the naming and decorating, but, as far as I can see, the airports are all named for the city that they are by (I'm sure there may be some exceptions to the rule, but I'm not that familiar with the smaller airfields)
From what I have seen (and bear in mind that I've not been here two months yet!), the famous person tends to have buildings / streets named after them, or statues put up, rather than airports.