Hey Sasha, it is very hard to reply to this question, as an Italian who speaks English with a good fluency I never bothered to think about how it would be to just speak English in Milan...
Finding Italian people with a native-like fluency it's very rare, but statistically Milan is the place with the highest concentration of good-enough English speakers in Italy - whether for tourism or for business, or, as in your case, for education.
If you stick with your university acquaintances and hopefully someone who speaks both English and Italian (even a little), you'll do fine.
I also hope that someone will manage accommodation details for you, because doing so on your own without speaking Italian would be difficult.
Since you have time to prepare, I would suggest to make sure that some vital details (accommodation, for example) are taken care of by someone who speaks Italian or can support you.
If you stop a bystander asking for direction, you have a 50% chance that he or she won't be able to speak English, especially among older people (it is definitely not like Scandinavian countries, where even granpas speaks fluent English).
But since you're coming in summer, with a lot more tourists around, chances of finding someone to speak to are better
Bottomline: yes, you can survive one month (or even more) especially if you surround yourself with other students and younger people (usually good English-speaking people move in "clusters", it is hard to randomly stumble in a good English speaker).
On your own everything will be a bit more difficult because unfortunately there are still huge part of the population that don't speak English at all, or they speak it very badly.