One contender would be Cook's Cottage, originally built in 1755, and removed to Australia and reconstructed in 1934. This has some association with the navigator, but he probably never lived there. It is now in the Fitzroy Gardens, in Melbourne.
Elizabeth Farm House in Parramatta was built in 1793 and is the oldest surviving building. There's a ruined stone fort built in 1629 by Dutch survivors of the Batavia wreck that is the oldest partially surviving structure.
The most famous building in Western Australia is probably the Round House in Fremantle, which originally served as the gaol.
See the related link below.
The Round House.
Kemp House
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kemp house
The Wallabies.
The Palace of the Governors.
Palace of the Governors
Kerikeri
Kemp house
The oldest city that is still lived in in the western hemisphere is believed to be Damascus, the current capital city of Syria, which is approximately 12000 years old.
New Zealand's oldest building is Kemp house, also known as the Kerikeri Mission House. It is located in kerikeri in the Bay of Islands in Northland.
The Virginia General Assembly which is the oldest legislature in the Western Hemisphere.
Kerikeri Mission Station, also known as KempHouse