Yeah, it's an idiomatic expression meaning 'all day' or 'from dawn till dusk'. Another idiomatic expression using 'dubh' is 'dubh le daoine' meaning 'crowded' or 'packed with people'. For example: bhí an áit dubh le daoine.
You're right, it means 'sick and tired in the 'fed up' sense. Ó dónaill translates 'tá mé dubh dóite de' as 'I am heartily sick of it'. It seems one use of 'dubh' is to intensify a statement in the way 'downright' would in English.