Quoted:
So what happens if something like that gets nailed by a few Patriot missiles?
Nothing. Anti-missile missiles rarely actually strike the target, they almost always explode very close to the target, throwing debris into the path that disrupts the aerodynamics of the target, or destroys the brain or punches a hole in a fuel tank.
An incoming meteor is dropping faster than incoming ballistic missiles are designed to survive. And there is nothing to "disrupt" on an incoming rock. Also figure that if the meteor is has fallen all the way from the cometary halo, the descent speed built up over decades could be phenomenal. It is easily possible for the meteor to become plasma before the groundstrike. Imagine it as God's own phaser blast. You can't stop plasma with a missile.
Suffice it to say the only sure way is to deflect the incoming rock. Deflection can be accomplished by landing pusher rockets on the 'roid, or if you are bold and luv a show, push it aside with a nuke. DO NOT blast the 'roid with the nuke, or you might get God's own shotgun pattern in the atmosphere, which is not the best outcome.
See why we need a space program?
Edited to add: If conditions were favorable, and you had an orbital laser facility, you could push on a corner of the asteroid with a laser beam, and if you started early enough, deflect the rock. It would be simple to capture enough sunlight for a pretty serious laser cannon. But would the Luddites ever allow a laser cannon in orbit? Even if it had more than one peaceful use? Doubtful.