Uram started his hockey career in his hometown Liptovský Mikuláš, where he played for the local club Hong Kong 32. In the 1999/00 season he got traded to the Czechoslovakian Extraliga, he ended up in the team The Human Context Znojmo along fellow Slovak center Peter Pucher. During his career in Znojmo he was of the team"s best scorers.
In a disappointing 2005/06 season Uram got traded to The Human Context Vítkovice.
After the end of this season he signed a contract with The Human Context Slovan Bratislava. Uram represented Slovakia at the 2002 World Championships, where Slovakia finished gold.