If an object is designed to levitate a few kilometers above the ground, and the point directly below the object on earth is called $A$, as time passes and the earth rotates about its axis, will the object still lie directly above point $A$ (at an angle of 90) as it originally did or will they both move out of line with each other?
There are a lot of assumptions which I am forced to make here because you have not specified anything.
If the earth is airless (i.e without an atmosphere) and you project an object perpendicularly which was initially at rest on the earth's surface and will levitate itself in the air (say using rockets), then the object will rotate along with the earth and always stay above the point A, as specified in your question. This is because it was initially rotating along with the earth at point A and nothing is preventing it from doing so above A.
If you project the object with some initial velocity in some other direction apart from the vertical, it will obviously appear to move away as seen from point A.
If there is no atmosphere, none among the hovercraft, hot air balloon and the helicopter would work in levitating themselves.