"I drink my coffee with one spoon of sugar."
Translation:Πίνω τον καφέ μου με ένα κουτάλι ζάχαρη.
What case is ένα κουτάλι ζάχαρη in this sentence? And what case is ζάχαρη?
Both are in the accusative. Ένα κουτάλι is in the accusative because it follows a preposition (this is the rule of thumb). Ζάχαρη is in the accusative because something like γεμάτο (με) ζάχαρη is implied.
Also, is it acceptable to say με μία ζάχαρη as in the colloquial "with one sugar"?
Unless you are offered a bowl with sugar cubes, which is pretty unusual itself, it's going to sound ungrammatical.