Who won the first best actor oscar while playing a part on a war movie?
Actually, the very first "Best Actor" Oscar ever awarded, which was for the years 1927/28 went to Emil Jannings for his performance in two films, one of which was The Last Command, arguably a "war film" as it centers on a man who was a former Imperial Russian General.
The Last Command is a 1928 silent film directed by Josef von Sternberg, and written by John F. Goodrich and Herman J. Mankiewicz from a story by Lajos Bíró. Star Emil Jannings won the very first Academy Award for Best Actor in a Leading Role for his performances in this film and The Way of All Flesh, the only year that multiple roles were considered. In 2006, the film was deemed "culturally, historically, or aesthetically significant" by the United States Library of Congress and selected for preservation in the National Film Registry. The supporting cast includes Evelyn Brent and William Powell.