Mr Barr, originally from Strabane, was killed in Sunset House pub in the north inner city, on 25 April.
The victim worked in the pub as a barman but was known to gardaÃ­ (Irish police) for alleged involvement with dissident republicanism.
A paramilitary-style funeral was held for Mr Barr in Strabane last month.
At the time of his death, detectives said that they were keeping an open mind about who was responsible for Mr Barr's murder but were not ruling out the possibility that the killing was linked to a violent feud between two criminal gangs.
The Kinahan - Hutch feud is believed to have claimed the lives of seven men.
The man who was arrested on Friday morning is in his late 20s.
He is being questioned at Bridewell Garda Station in Dublin.
Police investigating the murder of County Tyrone man Michael Barr, who was shot dead in Dublin two months ago, have arrested a man.