How long does a large impulse take to travel through a solid?

If I had a 1000km long steel rod and I pushed one end of it, when would someone standing at the other end observe movement? Would it travel exactly the speed of sound in that material, or is the speed of sound variable depending on the amplitude of the impulse (ie would it travel quicker)?

The speed of sound in a solid doesn't depend on the magnitude of the impulse.

The pressure ($$P$$) waves travel fastest and are unaffected by the size of the impulse. Shear waves ($$S$$) waves might also be created in a large body, depending on the nature of the impulse, they travel more slowly than the $$P$$ waves.

P waves and S waves

In the case of steel the atoms are well organized in a lattice, facilitating a directional chain-reaction of movement through the rood all the way to the end.

The individual atom works like a little pendulum (harmonic oscillator). An you might know that the frequency of a pendulum or a swing does not depend on the amplitude. Hence the use of pendulums in clocks.

So, within a reasonable range of initial impact the properties of the material and not the amplitude will decide the propagation velocity