He had a twin brother George and an older brother Hugh an deputy-lieutenant. Archibald and George Mosman spent some time growing sugarcane in the West Indies before arriving in Australia aboard the Civilian in 1828. The pair promptly started a business in Sydney, establishing a warehouse on George Street which they used to ship wool to Liverpool.
Around 1832, Archibald left the wool business and moved into whaling, operating a pair of whaling vessels out of Sirius Cove.
He died in 1863. He was buried in Randwick cemetery. Today, Sirius Cove is known as Mosman Bay and the land nearby is now the Sydney suburb of Mosman, both named after him.