"He drinks the cup of coffee."
Translation:Ólann sé an cupán caife.
Still struggling with genitive case; why is this wrong: "Ólann sé cupán an chaife"?
No — the structure X an/na Y, where X is a noun and Y is a genitive noun, also makes X definite.
So is there no way to say "an X of the Y" in Irish? "A bird in the hand"? "An ace in the hole"? "A man of the people"?
The dative needs to be used rather than the genitive for that structure — éan sa láimh (although the Irish version of the proverb uses éan i do dhorn [“a bird in your fist”]), ás sa pholl, fear den choitiantacht.
Yes, it is. Some nouns have identical forms for the nominative singular and the genitive singular.