Print this upside down and use the M600 change filament command to do the color change. The color changes occur at 1mm level heights so you do not need a fancy printer to make this.

I started with red filament up to 1mm and then switched over to black for the next 1mm layer and then switched to silver for the rest of the top. The bottom was printed in silver for contrast.

If you are using .2mm layers the color change needs to happen every 5th layer (.2 * 5=1mm) until done. Some slicers will do the automatic M600 code insertion. If the slicer you are using does not do this you will need to manually edit the gcode file.

I recommend coating the coaster with clear spray paint to seal up the surface. Water can seep into the gaps and lead to all sorts of crud build up.