You can't be a "member" of a Rite. You are a member of a Church, and in another sense, of a diocese.
Milanese Catholics who are members of the Archdiocese of Milan are Latin Catholics, members of the Latin Church, just the same as all other Western Catholics.
If you live in Milan it may be the case that your parish uses the Ambrosian Rite, but some also use the Roman.
The lack of use of the Ambrosian Rite does not mean that you are no longer a member of the Ambrosian Rite because, like I said, you can't be a "member" of a Rite.
Afaik, the only priests who can celebrate the Ambrosian Rite are priests of the Archdiocese of Milan.
I think they do have their own Breviary, but I am 99% sure that the only people who can use it to fulfill an obligation to pray the divine office are members of the Archdiocese of Milan. This includes, I'm pretty sure, people who are living in a different place for studies or whatever. I think I have heard of a Milanese priest who studies at CUA who celebrates the Ambrosian Rite right here in the USA.
However, if you happen to run across an Ambrosian Breviary, you can use it for your own private devotion all you want.