Police said four other people were injured outside the restaurant which is near the presidential palace.
Information Minister Daud Aweis condemned the “cowardly” attack.
The National Union of Somali Journalists (NUSOJ) said it was “deeply shaken” by the “senseless and brutal” killing of Nur, the director of the privately owned Somali Cable Television.
“His remarkable career and unwavering dedication have left an indelible mark on Somali journalism, inspiring countless young journalists to strive for excellence,” the union said.
Nur, a staunch advocate of press freedom, had previously faced threats, the Somali Journalists Syndicate (SJS), said.
In October last year, journalist Mohamed Isse Hassan was among more than 100 people killed in twin car bombings in Mogadishu.
A month earlier, a reporter with state-owned Somali National Television, Ahmed Mohamed Shukur, was by killed by a landmine near the capital.
Al-Shabab has been carrying out deadly attacks in Somalia in response to a large-scale military offensive that seeks to drive out the jihadists.
Somalia’s President Hassan Sheikh Mohamud took office in May last year promising an “all-out war” against the group.