I've stayed at the Hotel Gran Duca di York on 2 different trips. It's about a 5-minute walk to the Duomo and directly across the street from the Pinacoteca Ambrosiana if that is of interest to you.
I enjoyed seeing the Last Supper - book tickets from the official site although this is late in the process so they may be sold for your days. It took me under 30 minutes to walk from the hotel to the Last Supper location.
I did the roof of the Duomo in spite of a fear of heights. I bought the combo ticket so did the elevator to the roof (which does not go all the way to the highest part of the roof), the interior of the Duomo, the excavation part at the front of the Duomo. It also covered the Duomo Museums but I ran out of time so did not see them.
Sorry, no food recommendations. I'm not a foodie.