I don't know where you've already looked to find Patrick coming to New Zealand, but these are the sites I have saved in my bookmarks for such a purpose:
Beverley's transcriptions (http://homepages.ihug.co.nz/~ashleigh/
Family Search (http://www.familysearch.org/
) Click on the link "Australia and New Zealand", then "New Zealand, Immigration Passenger Lists, 1855-1973"
Denise & Peter's "Our Stuff" (http://freepages.genealogy.rootsweb.ancestry.com/~ourstuff/
Olwyn's "NZ Bound" (http://freepages.genealogy.rootsweb.ancestry.com/~nzbound/
Gavin's "Ships to NZ" (http://freepages.genealogy.rootsweb.ancestry.com/~shipstonz/
Corey's "Yesteryears" (http://www.yesteryears.co.nz/
Ancestors On Board (http://www.ancestorsonboard.com/
Public Record Office Victoria (http://www.prov.vic.gov.au/
Appendices to the Journals of the House of Representatives (http://atojs.natlib.govt.nz/
NZSG CD "Passenger Lists - Victoria, Australia Outwards to New Zealand 1852-1923" (http://www.genealogy.org.nz/Products_48.aspx?CategoryId=11&a...
These are in no particular order except the one I happen to have them listed in here.
I'd start with the Family Search passenger lists, then Denise & Peter's site.
Keep in mind that Patrick may not have come directly to New Zealand, but may have gone somewhere else first.
Do you know roughly when Patrick came to NZ? If not, his death registration should tell you, assuming the informant knew (and gave) the right answer.
Hope this helps. Keep asking questions - but please keep in mind the topic of this board; you may need to shift to another if the discussion is straying away from Southland.