Uram started his hockey career in his hometown Liptovský Mikuláš, where he played for the local club HK 32. In the 1999/00 season he got traded to the Czech Extraliga, he ended up in the team HC Znojmo along fellow Slovakian 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 HC Vítkovice. After the end of this season he signed a contract with HC Slovan Bratislava.