The suspect was arrested near the city of Patra, some 130 miles west of Athens, News.Az reports, citing foreign media.
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He initially opened fire at the social security office, wounding an employee, police said.
He then fled and later opened fire on the ground floor of a court building in another part of central Athens, wounding several people.
The head of Athens Judicial Employees Union, Stratis Dounias, said that the initial information indicated the man had shot at the floor inside one of the offices in the court building.
At least three female court employees were slightly wounded by ricocheting shotgun pellets. The motive for the shooting was unclear.
State broadcaster ERT said the gunman had reportedly left envelopes with documents after the shooting, saying those were the reasons for his actions.
Gun violence is relatively rare in Greece, where firearm ownership is tightly regulated.
