New Years day is a public holiday there. So yes, many places will be closed.
Indeed, many shops stay closed until the 6th, with winter sales starting again on the 7th of January.
This is reflected in a post on tripadvisor:
Restaraunt wise we were amazed at how many closed over New Year. I
expected some would be closed on the 1st but some were closed from the
31st till the 6th! We did struggle to find a place to eat a couple of
times and ended up having to go to one of the numerous self service
restaurants which are fine but not exactly what we were looking for on
and this recommendation on the same page:
I live in Milan and I personally do not reccomend to spend the new
year in the city.
Basically there is nothing to do.
Milan people usually go out of the city for new year's eve: to ski, on
the lake, in the coutryside, to visit other cities. The situation is
different in the other Italian cities, like Rome or Florence.
Of course many restaurant and clubs are closed, maybe is the worste
day to see Milan.
So long story short (unless you have specific sites in mind, in which case I'd check their individual websites), in general a lot of stuff is closed during this holiday period.