Original facetious answer involving a ghost cloth
Since the material
doesn't experience any force other than gravity
the cloth will fall through the post, through the planet, come out the other side and then fall again. It will oscillate about the planet's centre, exhibiting perfectly undamped simple harmonic motion, assuming the planet has constant density with radius.
You need the cloth's atoms to experience electromagnetic forces if you want it to be unable to pass through the post, and probably want the constituents of their nuclei to experience the strong nuclear force as well so they behave in a way that's at least somewhat reasonable...
Answer to corrected question
If the top of the post is larger than the cloth (so it's basically a table) and the cloth is dropped from a perfect horizontal alignment, it will maintain its horizontal alignment as it falls.
Ignoring quantum uncertainty in atomic positions & assuming the cloth has cooled to absolute zero given the lack of any other energy sources (including no cosmic microwave background at 3K) in this universe, so its atoms are not vibrating, every atom will make contact with the post at the same time and potentially bounce back directly upwards. Depending on the coefficient of elasticity between the post and the cloth's atoms, the cloth will rebound some height and then fall again. If the coefficient of elasticity is 1, the cloth will return to its original height before falling again and the bouncing will continue forever. If it's less than 1, the bouncing will gradually die away until the cloth's gravitational potential energy has been converted to heat in the post and cloth, which will then be radiated into the vacuum.
If we take into account quantum uncertainty and/or classical vibrations of atoms for the cloth not at absolute zero, roughly the same thing happens, except that there will be some differences between neighbouring atoms' touchdown times. However, all this chaos will only be apparent over extremely small scales (~1 atomic diameter) and so on the scale of the cloth itself would cancel out and give no significant difference in behaviour (ignoring the possibility of some fancy effects which can happen at or near absolute zero, like superfluid helium, since you haven't mentioned the chemical constitution of the cloth).
If the cloth lands on the edge of the post it will slide off and land on the ground, assuming this cloth doesn't experience friction. If it does experience friction then it may fold over the edge of the post and stay there or slide off, depending on whether product of the coefficient of static friction between the cloth and the post and the weight of the cloth on the post is greater than the weight of the cloth hanging over the edge.