Judge Jane Farish ruled earlier this year that he should stand trial across the Tasman.
She ruled that extradition would not be "unjust or oppressive'' to McGrath.
Legal aid and representation would be available to him in Australia, Judge Farish told him in June.
She also made sweeping suppression orders, after requests by The Commonwealth of Australia, to "ensure the integrity of trial'' and to mirror existing suppression orders in place in Australia.
But McGrath lodged an appeal to the High Court, and is there today in Christchurch with lawyer Phillip Allan and counsel Pip Hall QC to argue against the move before Justice Christian Whata.