A. The key might be the “continuously operating” part of the question. It would probably have to be a bar that somehow made it through Prohibition (1919-1932) with its doors open.
The first bar in Northeast may have been when the area was still called St. Anthony. Two French Canadians, Alexis and Theodose Cloutier, opened what they called a “bowling saloon” in 1850 at the intersection of Marshall and Dana Street (Fifth Avenue).
The new business ran afoul of a local temperance movement, and struggled to stay open in the next few years. There was actually a bowling alley of sorts, but the business was mainly aimed at giving men a place to knock one back, and the ambience was thick cigar smoke and spittoons…