Murati, who came to Austria in 2006,[2] made his first brief appearances for Swans Gmunden in the Austrian Basketball Bundesliga during the 2006/07 season and helped them win the championship title. In the following years, he developed into the team's key player. In 2010 and 2011, Murati was named the Bundesliga player who had developed the best in a single season by the internet portal eurobasket.com.[3][4] In 2010, he won the championship title again with Gmunden.
Murati spent the 2015/16 season with the Belgian first division team Leuven Bears.[5] He returned to Gmunden for the 2016/17 season.[6] In 2021, Murati was named Most Valuable Player of the Season after leading Gmunden to the championship title and averaging 18.9 points, 4.5 rebounds and 2.6 assists in 35 season games.[7][8] In the summer of 2021, he joined BC GGMT Vienna, which was described by his new club as a "sensational transfer".[9] He won the Austrian championship title and the cup competition with the team in the 2021/22 season.[10]
In 2014, Murati was in the Austrian national team squad for the first time after receiving Austrian citizenship the previous year.[12] He played a total of 42 international matches.[11] At the 2024 edition of the 3x3 Europe Cup, which took place on the Kaiserwiese in Vienna's Prater, he won the European Championship title together with Nico Kaltenbrunner, Toni Blazan and Fabio Söhnel.[13]