My question is does the shadow rotate around the object during day? or just change it's scale? or both?
Depends on what you want.
It is perhaps easiest to consider your "earth" flat, and the sun floating around that flat disc in 3D. (We discovered long ago that it's easier to compute orbits of other planetary rocks in the solar system when you assume the Sun is the center point, but you don't have that.)
For simplicity of discussion, let's assume the earth disc is in the XY plane, at z=0. Also, let's assume a day/night cycle is 24 hours at your earth. A shadow always points towards where the Sun is not, so it rotates 360 degrees in one day/night cycle.
Depending on the trajectory of the Sun relative to the disc, you get different effects.
If you let the Sun rotate in the XY plane at z=100 (above the disc), your world has 24/7 day, and the shadow is visible all 24 hours. If you let the Sun rotate around the disc in the XZ plane, at Y=0, your world has 12 hours day, and 12 hours night. The shadow still rotates 360 degrees in 24 hours, but is not visible during the night. If you tilt the Sun trajectory, or move its center away from the center of the disc, day and night cycles shift accordingly.
If you put the Sun so far away that its entire trajectory is at one side of earth (say, the XY plane, z=100, x of the center is 1,000,000,000), the shadow just moves back and forth between two angles.
I would expect that the Internet knows exactly what the trajectory of the Sun is relative to Earth in case you're interested, it's not very simple though, our disc rotates around itself in 24 hours, but we're tilted while we orbit around the Sun, which causes our seasons.