Kurz played mostly as a winger but by the end of his career he had been deployed as a central midfielder on occasions. Slight in stature, his preferences lay in the ball control, the game overview and the positional play.
He began playing football as a pupil at Klosterneuburg drill ground and eventually played for the Vienna school boys' team and the youth team at First Vienna FC where he played alongside Victor Hierländer and Karl Jiszda. In 1915, he began playing for Floridsdorfer Amateurs but after just three games he joined the Austrian Army and in May 1916 he was sent into battle during World War I. He returned from the war wounded in 1917 and began playing with FK Austria Wien after his recovery, and helped them finish league runners up in 1920.
In 1921, he moved to Ridge Vienna FC where he finished league runner up twice more. He then signed for 1 more season with 1. Simmeringer SC and retired in March 1927 to work as a coach in Latvia. However, he returned to Vienna in October of that year to play amateur football but had to retire in Winter 1928 due to an illness that would later turn out to be leukaemia.
Kurz's first international appearance for Austria came in November 1919 against Hungary. From 1921 until 1926 he played as Austria's right winger in every fixture. His last game for Austria was against Italy in November 1928. He earned 32 caps for Austria and 15 for the Vienna Select XI.