There's really no technological way to shut down the entire (US) Internet. What's been passed gives the President vast emergency powers, including the ability to order ISPs to shut down their systems on national security grounds.
The whole thing is a joke. Lieberman is a joke. The people who supported this are idiots. There is this absurd notion that someone could wage "cyberwarfare" against us, and therefore the government needs to be able to control it as though it were like any other battleground. It's completely absurd.
What I hope happens--and what I suspect would happen--is that some ISPs would refuse to comply, and many private citizens would set up their own large networks. And while ISPs are indemnified against civil damages incurred by being offline, what of all the companies that depend entirely on having an online presence? They wouldn't take this lying down. They'd sue in a heartbeat.
This strikes me as one of those "rainy day" laws that will never be used to its potential extremes, because it would be so obviously destructive no one would stand for it.
I'd be more concerned about it being used to take down Web sites that are critical of the government, or are doing investigative journalism into areas the government doesn't like. Shutting down the whole Internet is a non-starter, but it's very conceivable that this law would be used to stifle speech on national security grounds.
I don't know which way things would go if this made it up to the Supreme Court, either. They haven't been coming down on the side of free speech lately.