Born in Toronto, Ontario, the son of Jesse H. Arnup (1881–1965), a Methodist minister who was Moderator of the United Church of Canada from 1945 to 1946, and Ella Maud Leeson (1883–1966).

Arnup was named a King's Counsel in 1950 and practiced law with Mason, Foulds, Davidson, Carter & Kellock (now WeirFoulds LLP).

In 1968 he won a notable victory over another noted litigator, John Robinette, in the four-year-long Texas Gulf Sulphur case, at the time the longest civil trial in Canadian legal history.

[2] In 1949 Arnup played a role in the reorganization of legal education in Ontario in 1949, and he was a bencher (board member) of the Law Society of Upper Canada from 1951 to 1970.

[5] In 1989, he was made an Officer of the Order of Canada "for his contributions to his Church, to legal education in the Province of Ontario and for his work in the development of Osgoode Hall Law School".