I had about the same amount of time last summer and here is what I did (without a guide):
- Toured the Duomo, including going up to the roof (which was a highlight of my day)
- Wandered around the Galleria. Continued out to Piazza della Scala. Didn't have time to tour La Scala but would have liked to do so
- Bought lunch at Luini Panzerotti
- Walked up Via Dante
- Walked around the grounds of Sforza Castle. Sat on the lawn for a while enjoying the musicians entertaining others relaxing and eating lunch
- Visited San Maurizio al Monastero Maggiore.
- Saw the Last Supper
- Back to the train station and onward
There are other things I would have liked to see with more time (like the Science & Technology museum, the Basilica San Lorenzo, and more time at Sforza Castle), but I felt my time was well spent.