Translation:I have a green shirt from Duolingo.
Actually, we use "with" to specify the fact that the shirt has a Duolingo banner/whatever on it (for example). If that was your intention, think about how would you say in English that you have a shirt on which some cherries are painted. "I have a shirt with cherries (on it)". That is the same logic. "Am o cămașă verde cu Duolingo (pe ea)"