Coussa was ordained a priest of the Basilian Alepian Order on 20 December 1920 in Rome by Isaias Papadopoulos, titular bishop of Grazianopoli, assessor of the Sacred Congregation of the Oriental Church.
In late 1929 he left for Rome, where he was the delegate of the Melkite hierarchy in the commission for the preparatory studies for the codification of the Oriental canon law.
While in Rome he also assumed the position of professor of canon law at the Pontifical Roman Athenaeum from 1932 until 1936.
From 1936 until 1953 he served as professor of the Latin Code of Canon Law, at the Pontifical Institute "Utriusque Iuris", Rome.
On 3 March 1946 Father Coussa was appointed Secretary of Interpretation of the Code of Canon Law for the Roman Curia.
Coussa was consecrated bishop 16 April 1961 in the Sistine Chapel, by Pope John XXIII, assisted by Giovanni Mele, bishop of Lungro, for the Italo-Albanians of Continental Italy, by Giuseppe Perniciaro, titular bishop of Arbano, auxiliary and vicar general of the apostolic administrator of Piana degli Albanesi, by Archimandrite Théodore Minisci, higoumène of the Italo-Greek monastery of Grottaferrata, and by Archimandrite Ambroise Kassis, superior general of the Basilian Order of Aleppo.
Coussa died unexpectedly in Rome due to peritonitis caused by appendicitis on 29 July 1962, just as the Second Vatican Council was opening.