Bloom imagines young publicans-to-be "Coming up redheaded curates from the county Leitrim, rinsing empties and old man in the cellar." The picture is of hicks from the countryside scraping by in the big city, doing menial labor and gulping the leftover portions of customers' drinks.
Leitrim (pronounced LEE-trəm) is in the northern part of Connacht, just south of Donegal. It has the smallest population of any of Ireland's 32 counties. Gifford observes that it "seemed remote and agrarian to Dublin, and its inhabitants were regarded as country bumpkins." He also notes (without attribution) that "old man" refers to the remainder in a drinker's glass.