You can search New Zealand libraries holdings at
http://nzlc.natlib.govt.nz. If you search 'Southland Times' as a serial title, you will find a number of Auckland libraries. It appears that most only hold recent issues.
Details about Auckland City Libraries newspaper collection is online at
http://www.aucklandcitylibraries.com/aboutthelibraries/collections/News-and-Newspapers.aspx.