This major case in this area is called Youngstown v. Sawyer. Basically there are three types of powers that the President can have (other than those explicitly granted by the Constitution). 1) Those specifically granted to him by Congress, 2) those specifically denied to him by Congress, 3) those in the "twilight zone" (this is a complicated area of the law).
Congress can grant the President the power to shut down the internet because it is entitled to manage the flows of interstate commerce, a power that is, and was intended to be, extremely broad. The major cases in that area are United States v. Lopez, Gonzalez v. Raich, and United States v. Morrison.
Thus, if Congress passes the bill, the President constitutionally has the authority to shut down the internet. If they do not pass it, this will be considered as an implicit denial of presidential power in this area, and Obama will not be able to do it.
On the question of what constitutes a national emergency:
The US Constitution is one of the few in the world that does not grant "emergency powers." Thus, a national emergency would have to be defined by the actual bill in question. You would need to look to the wording of the bill to determine what constitutes a national emergency.