Uram began his career in his home town of Liptovsky Mikulas, where he went through all the youth teams of MHk 32 Liptovsky Mikulas. During the season 1993/94 he made his debut in the Slovak Extraliga. Until 2000, he played almost exclusively for his hometown club, only in the season 1997/98 he played for HK Poprad. During the 1999/2000 season, he joined with Peter Pucher in the Czech Extraliga for HC Orli Znojemští. There he became a prolific offensive player, who became the top scorer of the Extra Liga in the 2000 /01 season. At the end of the 2005 /06 Uram then moved to HC Vitkovice Steel and a short time later returned to Slovakia for the HC Slovan.
In the season 2006/ 07 he won the Slovak Championship with the HC Slovan and was honored as MVP of the playoffs. He also was charged with 43 goals and 41 assists in 67 games leading scorer the Extraliga. A year later, he won the championship again with Slovan Slovakia. In June 2009, Uram was taken from the MHC Martin under contract for which he completed in the following one and a half years, over 100 games in the Extraliga. In January 2011, shortly before the end of the transfer window, he returned to Slovan Bratislava.
Since September 2011, he has returned to his hometown club MHk 32 Liptovsky Mikulas under contract.
Uram played at the 2002 World Cup for the Slovak national team and won the world title with this. Five years later he was re-appointed in the selection and participated in the 2007 World Championships in Moscow and Mytischtschi, where he scored 2 goals and 3 assists.
Awards and achievements
- 2001 top scorer of the Czech Extraliga
- 2002 winning the World Cup
- 2007 winning the Slovak Championship
- 2007 top scorer of the Slovak Extraliga and elected to the All-Star team
- 2008 Profit of the Slovak Championship