Budai was born in Budapest and played as a midfielder and forward for Ferencvárosi TC, Honvéd and Hungary.

Other members of the team included Ferenc Puskás, Zoltán Czibor, Sándor Kocsis, József Bozsik and Nándor Hidegkuti.

In January 1949 when Hungary became a communist state, Ferencváros were deemed unsuitable to become an army or police club because of its right-wing and nationalist traditions.

Instead they were taken over by ÉDOSZ, a food workers union and their best players, including Budai, Czibor and Kocsis, were conscripted into the army team, Honvéd.

During the 1954 FIFA World Cup he played in the group stage in the 9–0 win against South Korea and in the semi-final against Uruguay.