Cuthbert's Coffee Shop on Fountainbridge will offer a buffet, popcorn and ice cream during the film showings, which are due to begin in January.
Owners Mark and Stef Grant also hope to run a children's classic movie club on Saturdays.
Mrs Grant said: "The curtains for the large windows are being made and it will all take a while - 108 metres. But we are looking at the third week in January as our premiere night.
"We are truly excited about this proposition as the shop has a connection with Connery that we wanted to build on. We do not wish to be too high-brow and will go with what the guests ask to view. There will be a majority voting on the night, but it will be a classic in its widest definition."
The coffee shop will also project subtitled films on to its wall during the day.
Cuthbert's opened a year ago in the building once occupied by St Cuthbert's Co-operative Society, where Sir Sean worked as a milkman in the 1940s.
The St Cuthbert's name, which came from the parish in which the Co-op was situated, was lost in 1981 when it merged with the Dalziel Society of Motherwell to become the Scottish Midland Co-operative Society - or Scotmid for short.
Mr Grant was keen to revive the name and open a business in Fountainbridge because his grandmother spent most of her life there.