He was later confirmed dead at a hospital, an official of the Tokyo Metropolitan Police said.
Media reports said that the man was a right-wing activist. Police were checking his identity, said the second official, who spoke on condition of anonymity, citing policy.
The man was carrying a protest letter to Fukuda urging him to handle foreign policy and the issue of the Yasukuni war shrine "firmly," according to the second official.
Healso was carrying another letter to the media calling on them to promote visits to the war shrine by the people of Japan, the Tokyo Police official said.
While critics call Yasukuni a symbol of Japan's past militarism, right-wing Japanese extremists support the shrine, which honors Japan's 2.5 million war dead, including wartime leaders executed for their World War II crimes.
Former Prime Minister Junichiro Koizumi infuriated China and South Korea with his repeated visits. Fukuda, a foreign policy dove, has foresworn visits to the shrine.
Shootings are relatively rare in Japan, which has strict controls on handguns.