You will have to wait until 2020-2021 to get access to the birth records from NSW. Meanwhile, Archibald Eric Price served in WWII. His records are in the National Archives at http://www.naa.gov.au . Enter his name in the box at top right and you'll find the 2 records. Neither has been digitised but you can apply to have them digitized ($$$). It's usually well worth it.
I'm assuming you haven't got any further with this. If you need any more help just holler.
His death isn't in the Ryerson Index that I can see.