Szuper began playing hockey in Hungary at an early age, but also showed a flair for the arts. He became an accomplished actor and pianist, but it was between the pipes that he began to achieve prominence. He starred with Ferencváros TC as a 16-year-old and named the top junior player in Hungary. He moved on to the Krefeld Pinguine of the Deutsche Eishockey Liga for the 1997–98 season, followed by stellar play in the 1998 World Championships in C Pool representing Hungary.
In spite of his success in Europe, Szuper was an unknown at the goalie position when he broke into the Ontario Hockey League in 1998 with the Ottawa 67s. But over the course of the 1998–99 season, Szuper excelled in net in spite of splitting time with Seamus Kotyk. The 67s advanced to the 1999 Memorial Cup as the host team, then defeated the Belleville Bulls in the semifinals to advance to the Memorial Cup championship against the Calgary Hitmen. Ottawa prevailed 7–6 in overtime, and the 18-year-old Szuper was being talked about as an NHL prospect.
Szuper failed to hear his name called in the 1999 NHL Entry Draft, so he returned to Ottawa and continued to star. Ottawa won the Eastern Division, but fell to Belleville in the conference semifinals.
Szuper was then drafted by the Calgary Flames in the 2000 NHL Entry Draft in the 4th round (#116 overall). Szuper would play the next three seasons with the Saint John Flames of the American Hockey League, earning a Calder Cup championship for the 2000–01 season.
During the 2002–03 season, Szuper was called up to the NHL with Calgary due to an injury on the big club. Szuper dressed for 9 games as a backup, but failed to appear in a game. He is the first Hungarian native to appear on an active NHL roster for a game, but does not appear in the NHL register since he never saw the ice.
After failing to stick with Calgary after the 2002–03 season, Szuper played the 2003–04 season with the Peoria Rivermen of the East Coast Hockey League. He lasted one year in Peoria before returning back to Hungary for the 2004–05 season; from there it was on to Asiago HC of Serie A.
Having played most of the 2006–07 season with the EV Duisburg Die Füchse of the DEL, Szuper is currently playing for Malmö Redhawks as they compete in the Kvalserien and attempt to keep their spot in the Elitserien for the 2007–08 season. He transferred in December 2007 to Hockey Club Junior Milano Vipers. In August 2008 he move to Alba Volán Székesfehérvár in his homeland.