I can't recommend specific Pilgrim accommodation, and I suspect no pilgrim trail actually traverses this city, but I do know that Milan is not the place to just arrive and find something on an ad hoc basis. I tried that 4 years ago: I arrived at the main station and expected to find numerous modestly-priced pensions in the immediate vicinity. There were none, and a tedious search for something suitable ensued. My experience suggests pre-booking using a accommodation booking website.
Of course, others may have found a plethora of choices during their own visit, so perhaps they can offer more optimistic information...