Peter,
I found a marriage between Richard Walker and Elizabeth Jane Kinnear. It was in 1859 (so the arrival date of 1855/6 in NZ seems incorrect). They married on 18.4.1859 in Knocknamuckly Church of Ireland I Seagoe parish, Co Armagh. Richard was 26, a bachelor, a farmer, and lived in the townland of Kernan. His father was John Walker, parish clerk. Elizabeth Jane Kinnear was 22, a spinster, occ farmer and lived in Crossmacaughley. Her father Joseph was also a farmer. Witnesses were John Walker and Joseph R Kinnear.
An 1859 marriage fits with an 1860 arrival in NZ.
Griffiths Valuation for 1864 lists John Walker (clerk) on plot 31 in Kernan, which was just under 5 acres. (There were several other Walker households nearby, including a different John Walker, so they all may be related). That whole area today has been built on and is now a residential estate in the new town of Craigavon. Kernan Park, Craigavon marks the approximate area.
Griffiths also lists Joseph Kinnear’s farm in Crossmacahilly (standardized spelling). Crossmacahilly is near Portadown, which Elizabeth’s obit said was her place of birth. Joseph’s farm was plot 1 which was just under 4 acres. Today that location is close to the Bluestone Rd, on the edge of Craigavon. Judging by a modern map, the access lane is gone suggesting the farmhouse has been demolished.
You can see where these properties were, using the maps on Griffiths Valuation. You can switch between the contemporaneous maps and a modern one using the slider bar on the top rh corner of the maps page:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch
You can then follow through to see what happened to those occupants, using the Valuation Revision records on the PRONI website. They go up to 1929:
https://www.nidirect.gov.uk/proni
There were no Kinnears in Crossmacahilly by the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Armagh/Kernan/Crossmacahilly/
There were 3 Walker households in Kernan in 1901. You should be able to work out if any are related to yours by using the Valuation Revision records to identify each one.
http://www.census.nationalarchives.ie/pages/1901/Armagh/Kernan/Kernan/
The suffix (clerk) is deleted from John Walker’s name in the Valuation Revision records around 1864, suggesting he no longer lived there – died perhaps - and had been replaced by another John Walker (presumably a relation).
Joseph Kinnear’s name in Crossmacahilly changes to “reps of” around 1895, which indicates he had died about that time.
Statutory birth registration only started in Ireland in 1864. Prior to that you need to rely on church baptism records. Assuming both families were Church of Ireland, there are 2 churches in Seagoe parish. Knocknamuckly (where the marriage took place) has records from 1838. Prior to that the family probably used Seagoe church. It has records from 1672 (with some gaps). Neither set of records is on-line yet, but there are copies in PRONI (the public record office) in Belfast. A personal visit is required to view them.