Do you have any other information about this person? marriage? death? children? parents?
Is it possible that the spelling of the surname could be different? eg BUDDEN

I had a look at the Victorian shipping records and only found two
BUDIN arrivals.

Adolphe BUDIN arrived on the "Gipsy Queen" 1855 aged 25 years.

Mr. BUDIN arrived on the "Moldavia" 1913 aged 34 years.

I am related to this line and am now investigating the budden connection which is scant. I believe it is linked ot an aboriginal heritage but the document i know exists is eluding me. Am happy to share further if you are interested :-)

Pick one;

Name: Henry Bowden
Birth: abt 1850
Departure: London, England
Arrival: 27 May 1885 - Brisbane

Name: Henry Boyden
Birth: abt 1851
Origin: British
Departure: Melbourne
Arrival: 5 Oct 1879 - Sydney, New South Wales

I can't find a Henry BUDIN at all these are about as close as I can get.
Name: Henry Boydon
Birth: abt 1852
Origin: Chatham
Departure: Hobart, Tasmania
Arrival: 18 Dec 1883 - Sydney, New South Wales

This one is close. Pity you don't have a birth or death date. Or a state that he may have landed in.
there are about thirty that sound like BUDIN

Name: Henry Budden
Birth: abt 1816
Origin: Portsea, Hants
Arrival: 16 Jan 1839