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.
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."
Gifford also notes, without attribution, that "old man" refers to the remainder in a drinker's glass, suggesting that the servers are swilling leftover portions of customers' drinks. Slote, on the other hand, quotes from an entry in Partridge's Dictionary of Slang that defines "old man" as "That part of the beer engine in which the surplus beer collects." Given the passage's reference to the cellar, Slote's reading is probably preferable.