The 25-year-old posted the videos on the social network site on Monday. In one of the videos he fired a gun and suggested he would kill himself and foreigners.
Police in the Bavarian capital investigated the video and searched for the man.
And at 4pm on Monday they received reports of a gun being fired on a field near a shop in Obersendling in the south of the city.
The description of the shooter matched the man they were looking for from the Facebook videos, police said on Tuesday.
Officers arrested him at his home on Monday evening. They found the weapon he was posing with which was in fact a stun gun.
He was taken to a psychiatric facility.
The man is now being investigated on suspicion of disturbing the peace. He has been investigated in the past for making far-right statements.
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