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.

Larger companies like Welsh National Opera (WNO) and National Theatre Wales (NTW) will see budgets fall.
Altogether, 67 organisations will share a funding pot of £27.5m.
ACW chairman Prof Dai Smith said it had been a "challenging" process but a third of organisations - mostly smaller arts groups - will not see any cut.
Opera, theatre and dance companies, as well as educational groups and arts centres, will be finding out about their proposed budgets for 2016/17.
The funding announcement is the final stage of ACW's investment review, the first in five years.
It has now decided how many companies should continue to receive regular core funding, and how much they deserve.
ACW has created a new Arts Portfolio Wales which sees 67 companies receiving funding for their core work, while two organisations receive lottery funding for specific projects and events.
Companies which receive less than £150,000 have been protected from any cuts.
The largest beneficiary of Arts Council funding, WNO, will see its funding fall from £4,539,538 in 2015/16 to £4,380,654 in 2016/17.
But the budgets of two organisations which receive annual grants have been increased. Ballet Cymru's funding rises from £193,842 to £243,842  while Sinfonia Cymru's budget almost doubles from £111,459 during the current financial year to £210,459 from April.
As well as distributing annual grants to fund the core work of Welsh arts companies, called revenue funding, ACW also distributes lottery funding for specific projects.
Prof Smith said the ACW was determined to be "bold, brave and far-reaching".
"In spite of a further £1.5m cut in Welsh government funding, we have been able to sustain a nationwide network of high-performing organisations.
"We have also protected them against the full impact of the cuts."
Of the companies that will receive funding from April, four companies will be members of the portfolio for the first time:
During the investment review a total of 94 companies applied for more than £32m from ACW. As well as four new companies, five others were told they would not continue to be funded from April 2016:
Police investigating the murder of County Tyrone man Michael Barr, who was shot dead in Dublin two months ago, have arrested a man.
[NEXT_CONCEPT]
Arts companies face cuts of up to 3.5% to the public money they will receive from April, the Arts Council of Wales (ACW) has announced.