National police said the officers were alerted early Friday morning that smoke was rising from the synagogue and came face to face with the man when they got there.
Authorities are trying to regain control of the capital of the French-ruled Pacific island territory, where rioting over electoral reform has resulted in four deaths and hundreds of arrests.
Senior official Kim Yo Jong said recent weapons tests were part of an arms buildup plan and not meant to advertise weapons for Russia’s potential use in Ukraine.
The shooting was the first major assassination attempt on a European political leader for more than 20 years, and has drawn international condemnation.