Tokyo police arrested Mitsuyoshi Kamiya, a resident of the southern island of Okinawa, for making the bomb threats from a pay phone in downtown Okinawa, a police spokesman said.
He said the suspect also threatened to bomb Camp Schwab, a US military base on the island.
Kamiya admitted to police that he made the bomb threats, but his motive was not immediately known, the police spokesman said. He did not provide any other details about Kamiya.
The island is home to about half of the 50,000 American troops in Japan, and residents have frequently complained about crime, noise and other issues related to the US bases.
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The US government wants to relocate one base, the Marine Corps Futenma air station, to another area of Okinawa, but many people want it moved completely off the island.
In three phone calls in March, Kamiya used threatening phrases in English such as "Bomb Camp Schwab" and "Bomb the embassy" which forced the embassy to increase its security, the police spokesman said on condition of anonymity because he was not authorized to speak to journalists who are not members of the police press club, which is largely limited to Japanese media.
State Department spokeswoman Jen Psaki yesterday said the US government takes threats to American diplomats seriously, and that it would not comment on specific details of any threats.
Kennedy, the daughter of the late President John F Kennedy, arrived in Tokyo in November 2013 as the first woman to serve as US ambassador to Japan.