In the pubs of Joyce's time, a "curate" was a bartender. The name comes from the Catholic church, and puns on the notion of being a servant or server.
The Catholic church delivers the cure animarum, the cure or care of souls. A curate is a person charged with this responsibility, so parish priests are often called curates, as in the French title curé. But in England and Ireland the term typically refers to priests who assist the parish priest—junior members of the profession, assistants to the celebrants of the Mass.
In Calypso Bloom imagines Larry O'Rourke "watching the aproned curate swab up with mop and bucket," and then thinks of these employees "Coming up redheaded curates from the county Leitrim, rinsing empties and old man in the cellar. Then, lo and behold, they blossom out as Adam Findlaters or Dan Tallons." The analogy between a clerical vocation and a career in alcohol-delivery is clear: pay your dues, faithfully serve the mother-institution, and before too long you can have your own establishment.
In Lestrygonians Bloom is respectfully addressed as "sir" by the "curate" in Davy Byrne's, who serves Bloom his cheese sandwich and glass of burgundy. Davy Byrne himself chats with his customers, but does not do the serving.