They would say, of course, that because nothing can be allowed to transmit faster than light, it would take 8.20 minutes for the fellow at the other end of the rod to feel the push.
But wouldn't the work have already been done? The rod is there. It's the same concept for why gravity can 'escape' a black hole.
If we set up this experiment, when the rod was pushed forward, it seems to me the fellow at the other end would immediately feel it.
Do all the atoms in the rod really have to compress and transfer all the way to the other end? To avoid faster-than-light information transference?
EDIT @planetmaker has just pointed out it is far, far worse than I thought. The push would actually take an entire year to feel since apparently it would move at the speed of sound.