Koschatzky attended the Realgymnasiums Bundeserziehungsanstalt Liebenau, in Graz, until 1936 when he moved to the Military Middle School, also in Liebenau. Between 1940 and 1945 he served in the German armed forces.
In 1945 he commenced studies in art history, archeology, history and philosophy at the University of Graz and graduated in 1952. He had financed his studies with a jazz ensemble, working for the British military broadcasters (britischen Besatzungsmacht betriebene Sendergruppe Alpenland) and the Styrian Provincial Theatre.
Koschatzky married Trude Caroline Bauer in March 1948; she died in July 1994. On 15 July 1996, he married Gabriela Elias, an Austrian author, museum curator and cultural journalist.
In 1953, Koschatzky joined the Office of the Styrian Government, and moved in 1955 to the Neue Galerie at the Universalmuseum Joanneum in Graz where he worked from 1956 to 1961 as a director. He produced acclaimed art exhibitions, art discussions, lecture series on the radio, art walks in Graz, art tours and activities in public education; he was an eloquent expert on the arts. In 1961, he succeeded Otto Benesch as director of the Albertina ago as his successor.
From 1 January 1962 to 1 October 1986 he was appointed, by Minister of Education Henry Drimmel, as director of the collection of Prints and Drawings at the Albertina in Vienna. During his term, there were more than 200 exhibitions; Koschatzky was one of the most famous directors of museums in Vienna. The list of exhibitions held during his term as director began in 1962 with Gustav Klimt (celebrating Klimt's 100th birthday). Koschatzky curated 100 exhibitions outside the Albertina, including important ones in foreign countries. In 1972, there were exhibitions of important drawings from the Albertina in Moscow and Leningrad; in 1985 in Washington DC and in New York City. In 1986, Koschatzky presented European draftsmanship in Beijing. He continued to produce exhibitions after he had retired.
Koschatzky participated in television programmes about artists.
Between 1973 and 1989, he taught at the Universities of Vienna and Salzburg.
