Civil registration of births in Ireland didn't begin until 1864 so there will be no birth certificate for James Jones born c1838 in Armagh (County or City?).
You would probably need to search more records in Australia(?) before looking in Ireland.
For pre-registration information in Ireland you need to look for church records if they still exist. To do so you need to know the religion and where they lived (parish if not townland) not just a county. Occupations can also help find clues.
