Welcome to Rootschat. You've been really lucky to have so much information to start with from Australian records but unfortunately Irish research isn't that easy.
Registration of births, deaths and Catholic marriages started in 1864 (other marriages from 1845) and first complete census is 1901. For pre-registration dates (such as James and his wife) you need to search for existing church records from that period (if any) but to do so you need to know both the family's religion and where they lived. It may be that your ancestors came from Armagh city or somewhere in the county.
If you were to find church records of a baptism for Eliseb (Elisabeth?) around 1854 with a father named Thomas and mother Rachel that would be brilliant (although many baptismal records only record the father's name) but James' baptism is another matter. With a common name like 'James Hamilton' there could be many born around 1843 and without even a father's Christian name to confirm this you won't know when you've found the right record.
As far as 'parents unknown' on James' records- death certificate depends on how much information informant is able to provide and is he gave same on his marriage record it could be that he was an orphan from an early age or just didn't know his parents' Christian names.
