Asklemmy

Your push would travel at the speed of sound in the stick. You could think of hitting a pipe with a hammer, the sound of the hit would travel at the speed of sound, same is true for you pushing the stick.

Kosmiczny Kij

I don't see this mentioned in any of the other comments: the repulsion between atoms that causes the movement to propagate through the stick is actually communicated via photons. So your push really generates the same kind of particles that your light torch is generating, and they travel at the same speed. Except in the stick it is slowed down by repeated absorption and excitation by the electrons of the atoms.

why wouldn't this work

because bullets are faster than whatever the fuck speed stickman is achieving
and even bullets are slower than light

That's what he meant by we'll use sticks on the other side

Nah, I prefer using quantum spookiness for that. Send a steady stream of entangled particles to the other person on the moon first. Any time you do something to the particles on Earth, the ones on the Moon are affected also. The catch is that this disentangles them, so you have only a few limited uses. This is why you want a constant stream of them being entangled.