I checked Find My Past for the record of John Fitzgerald's baptism in 1847. The address is transcribed as Beaufield.
Roots Ireland (subscription site) has the address as Beanfield. The name of the male sponsor is transcribed as Joice.
I could not locate any area that matches either Beaufield or Beanfield but perhaps somebody with more local knowledge will jump in here.
I then checked for baptism records of any more children to the couple and found just one.
William: 3 Jan 1850. You can view the original record on the parish registers here.
I did not find any other baptisms in Cork that named Catherine Lynch as the mother.
I found the record of the marriage of John and Catherine on 8 Feb 1846 in Youghal. The grooms address is transcribed as Clonpriest on Roots Ireland.
You can find that record on the parish registers also, use the link above.
I noticed that there is a marriage of a William Fitzgerald on 3 Feb 1846 with the same address and one of the witnesses is James Fitzgerald. These are likely relatives of John, possibly brothers or cousins!
Griffith's Valuation shows some records of a John Fitzgerald in Youghal in 1853 but it is difficult to know if it is the same John.
This list of Tithe Defaulters in 1831 shows 3 Fitzgerald entries for Clonpriest. Since John would most likely have been too young to be listed considering he married in 1846, one of these Fitzgeralds is likely to be his father.
You could try searching for an obituary for John in Australia which may give more detail. I presume you know the date he died. The website Trove is a free site run by the National Library of Australia and finger crossed you may find something there.