Assuming a non-maneuvering target, i.e., a satellite without thrusters, you just have to figure out when he'll be at (x,y,z) spot over the ground at some time(+/- t... that's your error), then launch some explosive body towards that point. Kablooie. No need to have a precise hit.
The news reports I have seen have said it was a Hit-to-Kill ASAT, meaning that it did not have a warhead, it hit the target to destroy it. It was the premise behind our ASAT in the 80's, the impact from the seeker hitting the satellite, basically two objects going in opposite directions hitting each other going about 17,000 mph or more, is enough to obliterate both objects. That is not easy to do, by any means. The Soviets were the ones who did the proximity warhead ASAT, it was supposedly was like a really big grenade going off near the satellite.
The Hit-to-Kill principle is also behind the Exo-Atmospheric Kinetic Kill Vehicles (EKV's) that MDA uses on the Ground Based Interceptors (GBI's) http://www.globalsecurity.org/space/systems/gbi.htm. The kinetic part of it is why many scientists and others don't like the MDA system, it is like trying to shoot down one bullet with another and you are not sure where the bullet you are trying to shoot down is going. At the very least, it is an extremely hard problem to solve even with all the money MDA gets, even they admit that. If the GBI's were nuclear tipped, the whole kinetic concept would be a moot point. It would also make it cheaper and more reliable, the Russians' ABM system supposedly uses nuclear-tipped missiles: http://www.missilethreat.com/missiledefensesystems/id.22/system_detail.asp ......a little ironic, isn't it?