For a NEO to be of concern it's going to have to be either very big or very hard (most others will just burn-up or be of no real consequence - just evacuate the strike zone). Very big and very hard are going to be a right bugger to smash up into small enough chunks to pose little-to-no risk.
You are going to have to coordinate a lot of rockets carrying a lot of bang to arrive at the right time and strike in a predetermined manner (all of which will be compounded by the target moving like absolute buggery and tumbling whilst it does so).
So one is back to softy-softly, gravity-well-catchy-monkey. You don't have to move it out to beyond Mars or something, just stop it from passing through a "keyhole" (where Earth's gravity *will* cause it to impact at a future time; and even then you'll have a few years to try again).