Budaj first played for Slovakia in 1997 at an “Olympic Hopeful” Under-16 tournament. He was a member of Team Slovakia at the 2001 and 2002 World Junior Championships and the 2000 World Under-18 Championship.
He made his Olympic debut at the 2006 Winter Games, posting a 2-1-0 record with a 2.01 goals-against average and .924 save percentage. Slovakia’s goaltender trio of Budai, Jan Lasek, and Karol Krizan ranked second only to Finland for highest save percentage (.937) through six tournament games.
Budaj appeared in one game for Slovakia at the 2008 World Championship.