“Couldn’t we just shoot it down with a missile before it activates?”
An EMP attack detonates at high altitude, but it would still have to reenter the atmosphere to hit the “sweet spot”.
Now, as to whether we could shoot it down, the Russians developed a plan for an EMP attack which they shared with NK. This plan involved attacking from a cross-polar trajectory, in which the warhead would approach us from the Southern hemisphere, and we have no early warning system in that direction. You can’t shoot down what you can’t see.
I don't think that it would have to re-enter the atmosphere. EMP devices can be detonated at an altiude of 40 - 400 km. Low Earth Orbit range is 200 - 2,000 km with the lower range set by the rate of orbital decay. So you could have an on-station EMP device orbiting in the range of 200 to 400 km that would not require de-orbit or re-entry.