Am getting nowhere with my own research, so I thought I'd procrastinate and look at your request again...
1. Eastern Bush School's admission, progress and withdrawal registers for 1892 – 1928 are held by Invercargill City Libraries and Archives. If you do want to find out the names of the pupils, this would be your best bet.
More details including contacts here:
http://thecommunityarchive.org.nz/node/73308/summary2. Have a read through "Papers Past" - its a great online collection of old New Zealand newspapers.
http://paperspast.natlib.govt.nz/cgi-bin/paperspastThe area you are looking for is covered by the Otautau Standard and Wallace County Chronicle (1905 - 1932). Just a quick look through myself I could find out:
The school was built in 1891, to accommodate 50 pupils. Only ever a sole teacher, and here are the ones I could establish (again, you could probably rule out the years where the head teacher was female):
1892 – 1893 Mr Meiklejohn
1893? – 1898 Mr Barraclough
1899 – 1900 Mr John McFadden
1901 - Miss Taylor
1902 – 1910 Miss Lena Fahey
1911 - 1912 – Mr R M McIntyre
1925 – 1928 Miss Leila Rout
1928 - 1930 – Mary E Young
1931 - 1932 – Mr Ernest A Boon (there were 16 pupils in 1932 - so similar in number to your photo)
Some big gaps there, but it might help. A number of the articles also include lists of school pupils participating in school picnics etc.
Good luck.
Heather