Ude started gymnastics at the age of six after being recruited by a coach at his school.^[2] He won a silver medal on the floor exercise at the 2004 Junior European Championships.^[3] He finished sixth on both the floor exercise and the pommel horse at the 2006 European Championships.^[4]

At the 2007 European Championships, Ude finished seventh in the pommel horse final.^[5] He advanced into the all-around final and the 2007 World Championships, finishing 22nd and earning a berth to the 2008 Summer Olympics.^[6] He won a silver medal on the pommel horse at the 2008 European Championships, behind Hungary's Krisztián Berki.^[7] He then represented Croatia at the 2008 Summer Olympics and advanced to the pommel horse final, where he won the silver medal after winning an execution score tiebreaker over Louis Smith.^[8][9] He became Croatia's first Olympic medalist in gymnastics.^[10][11]

Ude won a silver medal on the floor exercise at the 2009 Mediterranean Games, behind Eleftherios Kosmidis.^[12] He had surgery on his shoulder in September 2009 and did not compete at the 2009 World Championships.^[13] He advanced into the pommel horse final at the 2010 World Championships, finishing fifth.^[14] He then finished 19th in the all-around final at the 2011 European Championships.^[15]

Ude placed 16th in the all-around at the 2012 Olympic Test Event and qualified to represent Croatia at the 2012 Summer Olympics.^[16][17] There, he only competed on the pommel horse but fell and placed 32nd in the qualifications.^[18] He did advance into the pommel horse final at the 2013 European Championships and finished eighth.^[19] He won the pommel horse title at the 2014 Osijek World Challenge Cup.^[20] At the 2014 World Championships, he won a silver medal in the pommel horse final, behind Krisztián Berki.^[21]

Ude finished 33rd in the all-around qualifications at the 2015 World Championships, making him the fourth reserve for the final. As a result, he advanced to the 2016 Olympic Test Event. There, he finished 15th in the all-around and earned a berth to the 2016 Summer Olympics.^[22][23] At the Olympics, he only competed on the pommel horse but fell in the qualifications and did not advance into the final.^[24] He was the first reserve for the pommel horse final at the 2017 World Championships.^[25] He missed the 2019 World Championships due to a knee injury that required surgery.^[13]

At the 2020 European Championships, Ude won the second European medal of his career– a silver on the pommel horse, behind Matvei Petrov by 0.033 points.^[26][27] He won the pommel horse bronze medals at the 2021 World Challenge Cup stages in Koper and Mersin.^[28][29] He then competed at the 2021 World Championships and qualified for the final, but he fell and finished seventh.^[30]

Ude won the pommel horse title at the 2022 Cottbus World Cup.^[31] He then finished seventh in the pommel horse final at the 2022 European Championships after a mistake that threw off the rhythm of the routine.^[32] He also made the final at the 2022 World Championships but fell and finished eighth in an attempt to increase his difficulty score.^[33] He won a silver medal on the pommel horse at the 2024 Osijek World Challenge Cup.^[34]