I guess I wouldn't do much since if it the internet would never come back, there'd be no use trying to salvage anything. If I was given one hour, I presume I wouldn't be thinking straight and so, I'd leave it at that!
The whole design of the internet virtually prohibits its "going down."
It wasn't initially designed for what people now call the "web." It was designed to provide robust and reliable connectivity between military, scientific, research and university computers, and was initially funded and developed by the Defense Advanced Research Projects Agency in the 1960s. Its core design paradigm is redundancy and fault tolerance, and doesn't rely on any one computer or connection. Each message is broken into several packets and sent out using arbitrary connections through arbitrary computers.
You can destroy most of it and as long as there is still some connection path somewhere between the source and destination, the message will get through. Even if the message is interrupted, or parts of it mangled, systems will retry delivering the mangled packets until the entire message gets through.