Amissah-Arthur was born on 29 April 1951 in Cape Coast, Ghana. He attended Mfantsipim School and later graduated from the University of Ghana with a B.Sc. and M.Sc. in Economics.

Amissah-Arthur served as Deputy Secretary for Finance and later as Deputy Minister for Finance under Jerry Rawlings from 1986 to 1997.

He was married to Matilda Amissah-Arthur and had two children. Amissah-Arthur was a supporter of Accra Hearts of Oak football club and enjoyed playing table tennis.

He was a lecturer at the University of Ghana and the State College of Education in Nigeria. He also worked as a research assistant at the Institute of Statistical, Social and Economic Research.

He was appointed Governor of the Bank of Ghana in October 2009 by President John Atta Mills, serving until August 2012.

Amissah-Arthur became Vice-President under President John Dramani Mahama in August 2012, following the death of President John Atta Mills. He served in this role until January 2017.

Amissah-Arthur died on 29 June 2018. He was accorded a state funeral, and several institutions have been named in his honour, including the Amissah-Arthur Senior High School in Moree.