If I pull a long stick one meter towards me, how long will it take before the other end is displaced one meter?

The speed of sound is the speed at which disturbances travel through the medium. When you pull at one end of the stick, you are essentially inducing such a perturbation. Microscopically, this is due to the fact that molecules along the stick react to this only by interactions with its neighbors, and have no idea that you've pulled at the stick until that disturbance reaches its neighbors.

Edit: regarding your edit, changes in pressure correspond to changes in microscopic position. What you're asking about is essentially a macroscopic manifestation of the same phenomenon. To hone your intuition, imagine a 10-foot wide steel cable, the kind that supports a bridge. If an earthquake shakes the whole system, the cable is going to undergo some oscillatory motion. On one hand, these waves travel through the steel cable like some kind of sound wave, but on the other hand, the oscillations might be macroscopically visible and would bear no distinction from the result of a friend shaking one end of the cable really hard.

It will be the speed of longitudinal elastic waves in the rod: $$ v_{rm longitudinal}= sqrt{frac Yrho} $$ where $Y$ is Young's constant, and $rho$ is the mass density of the material in the rod. I usually call this the "speed of sound" in the rod, although it's not exactly "sound."

About $0.25$s.

There are two components in this question:

• Purely physical - how quickly the other end of the stick responds to a pull, and the answers given above are correct (the excitation travels through the stick with the speed of sound - rather similar to how visual information propagates with the speed of light)
• Human - how quickly the person holding the stick will feel that the stick is being pulled. Human reaction time is usually taken to be $0.25$s. In this case this time is much longer than the time it takes for the sound wave to travel through $1$m.

Update
My initial answer was for a 1 meter long stick. SInce then teh questionw as edited: so, if the stick is sufficiently long, we do need to account for the speed of sound: $$ 0.25 + frac{L}{c_s} $$

this would mean that somewhere between me and her there is a difference of one meter,

Not necessarily. I don't know what material your stick is made of, but the speed of sound in solid materials can be multiple thousands of meters per second. The only way there could be a "difference of one meter" is if you manage to move your end of the stick a full meter in less time than it takes for the "news" that your end is moving to reach the other end. That's maybe only a few tens of milliseconds. You would have to pull extremely hard in order to move your end that fast.

which means that the stick will be elongated (or break).

Yes. If you pull hard enough on your end, you can break the stick. Doesn't matter what it's made of. Every material has some finite tensile strength.

As for "elongated," then that's a yes even if you pull very gently. There is no such thing as a perfectly rigid body.

You seem to believe that when you pull the stick through one meter, momentarily the stick is somehow stretched by one meter at your end before the other end moves to restore the stick's length, which is not the case.

When you start to pull the stick it will elongate microscopically (if the stick was replaced by something more elastic, like a weak spring, you would see it extend).

The effect of the microscopic extension will propagate along the stick at the speed of sound in the stick until a position of equilibrium is reached at which point the force you are applying to the stick will be felt by your friend.

Depending upon the material from which the stick is made, your friend might feel the pull almost instantaneously. The speed of sound in a wooden stick would be around 3 to 4 kilometres per second, so it would travel through a 100m stick in perhaps 0.03s

From that moment on, the stick and your friend are moving in unison (more or less) through the 1m.

Something of the reverse of what I have just described will happen when you apply a force to bring your end of the stick to rest after moving it 1m. The stick will compress somewhat until it and your friend are at rest.