There is a entry on RYERSON for the Sydney Morning Herald FUNERAL Notices of 12 April 1893 for a James McCORMACK Funeral that day.
You could access that via TROVE.
The 10 May 1893 James McCORMACK is also on RYERSON, from a DEATH notice.
http://srwww.records.nsw.gov.au/indexsearch/keyname.aspx
there is a record here, possibly him, under Intestate Estate Case Papers. 1893, James McCormick aka McCormack, died "The Hospice, Sydney", a clerk, native of Ireland.
This could refer to Sacred Heart Hospice, Sydney? His death appears to be listed under 'McCormack' on NSW bmd, father William, mother unknown
'The Hospice' indicates he was ill rather than there from an accident etc.
Given he was 19 and it was 1893, possibility he was suffering from TB?
Agree with jorose, probably Sacred Heart Hospice, Darlinghurst, Sydney.

Dawn M