It was not immediately clear whether the two incidents were related, but reports said that the suspect in the hospital case had sped away on a motorbike.
Images on television showed the post office suspect, reported to be possibly in his 80s, inside the building in a baseball cap and a white shirt under a dark coat, with what looked like a gun attached to a cord around his neck.
"At approximately 2:15 pm today (0515 GMT), a person took hostages and holed up at a post office in Chuo 5-chome area of Warabi city... The perpetrator is possessing what appears to be a gun," the city's authorities said on their website.
"Citizens near the scene are urged to follow police instructions and evacuate in accordance with police instructions."
Local media NTV reported that at least one woman in her 20s and another woman in her 30s were inside the post office, citing police sources.
Police were talking to the man by telephone, NTV reported.
The Yomiuri daily had reported earlier that around 10 post office staff may be inside the building and that the man may be carrying kerosene.
Police urged 300 residents in the nearby area to evacuate, broadcaster TBS said.
Several police cars with flashing lights were situated around the substantial three-storey building as night fell, television footage showed.
The surrounding streets were deserted.
- Hospital shooting -
The incident came as police investigated a suspected shooting at a hospital in nearby Toda, on the outskirts of Tokyo, earlier in the day.
Two people were slightly wounded -- reportedly a doctor and a patient -- after shots were apparently fired from the street into a ground-floor room in the hospital.
"After 1 pm, I heard a woman shouting 'Someone, please come,' and a nurse told me, 'Stay away from the windows and keep your head low,'" a man in his 60s who was inside the hospital told broadcaster NHK.
"Around 2 pm, I looked inside the doctor's office, and saw a pool of blood next to an examination table. I didn't hear a gunshot. But a nurse said she heard two gunshots," the man said.
Violent crime is rare in Japan, in part because of strict regulations on gun ownership.
Japan has one of the lowest murder rates in the world.
But recent years have seen violent crimes, including gun attacks, make headlines, most notably the assassination of former prime minister Shinzo Abe last year.
In April, a man was arrested for allegedly hurling an explosive towards Prime Minister Fumio Kishida as he campaigned in the city of Wakayama. Kishida was unharmed.
The following month, a man holed up in a building after allegedly killing four people, including two police officers and an elderly woman, in a gun and knife attack.
Masanori Aoki, 31, was taken into custody at his house near the city of Nakano, police said at the time.