I think I just don't understand American constitutional law all that well, but in what way is the first amendment being violated? To my understanding the first amendment is freedom of speech and expression, but does not in any way guarantee someones right to be anonymous while making that speech. This bill does not stop people from speaking, only that they do so in a personal capacity. So basically you can say exactly the same thing on the internet as if you were out on the street and saying it in public yourself. However, like I said that is just my interpretation of this.
That said though, this is a stupid plan. This will be almost impossible to implement as the internet stretches far beyond state boundaries. People will just stop using new york based sites, and may strongly harm any internet related business of those sites. There are many ways to deal with cyber bullying, but this is not one of them.