Q:

How quickly is momentum transferred through an object?

If there was a pole, that was a light year long, out in space. (Away from all physical interference.)
If we pushed one end of it, would the other end move instantly? And does it's speed depend on what it's made of?

A:

By pushing one end of the pole you would send a compression pulse down the pole which would travel at a finite speed much less than the speed of light and so it would take many years for the pulse to reach the other end of the pole.  
If the pole was made of steel then the speed of the pulse would be approximately $8000\,\rm m\,s^{-1}$ which is the speed of sound in steel.  
The experiment which is described here to measure the speed of sound in a metal rod is based on the idea of a pulse travelling down a rod.

