The embassy, in an advisory obtained by Reuters on Wednesday, said Tuesday's shooting took place in broad daylight in an area with busy traffic.
A British diplomat told Reuters the victim was a German national but German embassy officials were not immediately available for comment.
The advisory said that last month another European tourist was shot and seriously injured near a Muscat hospital in similar circumstances. Omani authorities said that incident was a hunting accident, it added.
"Both victims were alone at the time of the shootings. Neither was an obvious target for attack," the embassy said.
"It is difficult to be prescriptive over the right advice in cases of random shooting, but the community will wish to use common sense as they move around Muscat. It is important that the community not become unduly alarmed."
Oman, the second largest country in the Arabian Peninsula after Saudi Arabia, draws some 650,000 tourists a year with its tropical beaches, as well as ancient courtyards and palaces.
The independent oil producer has strong military and diplomatic ties with the West -- mainly Britain -- and its society is more tolerant of Westerners than some countries in the mainly conservative, Muslim Gulf region.