So, I saw this meme about some minister commenting that the Earth doesn't rotate, for if it did then planes only need to be in the air and the destination city would come to it. I know this is absurd but it got me pondering about how to achieve this practically.
I know that the ISS in a LEO covers the entire earth in 90 minutes. But how high do you need to go in a straight line so as not to be affected by the Earth's rotation and cities would literally come to you?
The Earth "compells" an aircraft to rotate with it through the fluid drag of its atmosphere. So a practical answer to your question is then "above the atmosphere", which is at about a $100{\rm km}$ height. This is the von Kármán line, which is often taken as the definition of the edge of space. The definition is made because at this height, a standard aircraft would need to fly at the orbital velocity to gain a lift equal to its weight. Therefore, the atmosphere above this height has a negligible effect on a spacecraft's dynamics and it will continue to orbit the Earth with negligible thrust indefinitely, if it flies at the orbital speed of about $7.2{\rm km\,s^{-1}}$ or more (in which case it will either undertake elliptical orbits with 100km or so perigee, or its orbit can be converted, through a sequence of small thrust manoeuvres, to a higher altitude circular one).
So one can definitely get to an altitude where one is not dragged along by the Earth's atmosphere, but one cannot hover there without continual thrust. Most likely one is orbiting, and, in low Earth orbit, the orbit speed is greater than the speed of rotation of the Earth's surface.
I don't know exactly what von Kármán had in mind as the standard aircraft, but chances are that the definition would work out pretty much the same for a very wide variety of aircraft.
As soon as you get above the atmosphere (about 100 km off the surface of Earth, give or take), then there's nothing in particular that compels you to follow the Earth's rotation.
Of course, once you get there, you will probably already be moving to some degree, depending on which mechanism you use to get yourself up. If you do, however, you can bring along a rocket engine and bring yourself to a complete stop with respect to the center of the Earth. The Earth will then be rotating under you.
However, you will then need to keep burning that rocket engine in order to keep the gravity of Earth from pulling you back into the atmosphere. Once you're out of fuel, you're going to fall.
It doesn't depend on the height.
Now you are on the rotating Earth, so you rotate with it (around the axe of Earth's rotation). Its speed is between 0 (on the poles) and around 1.5 Mach on the Equator (1 Mach = the sound of speed).
If you want to compensate this rotation, there are many ways, for example, you can simply sit on an airplane capable to go with this speed, and flying in the opposite direction. Concordes could to that, current passanger airplanes can do this only from around $55^\circ$ of the Equator. On them, the Sun went into the opposite direction on the Sky and you arrived earlier (in the local time) as you started the voyage. They flight faster as the Earth rotated below them.
But the ISS does not this. The ISS is on orbit, with around 25 Mach. Actually, it is going much faster as it would be needed to compensate the Earth's rotation. And it is going into around the same direction as the Earth rotates (not perfectly, there is around a $30^\circ$ angle between its orbit and the equator). It is done so, because this speed is needed to compensate the Earth's gravity by centrifugal force. If it would be slower (it had been much cheaper), it would fall.
