James Joyce would enjoy the irony: his museum really belongs to friend and fellow literary great, Italo Svevo, housing a significant collection of the Triestini's first editions, photos and other memorabilia. Joyce, who lived in Trieste from 1904 to 1920, is dealt with ephemerally, with a wall map of his haunts and homes and a Bloomsday bash in June (Svevo's birthday is also celebrated, on 19 December).
There's an interesting English video of Joyce's time in Trieste (ask the staff to play it). Staff can also give you suggestions for a DIY walking tour of Joyce sites.