De-gloss the surface of the faux-ceramic tile, using the power sander and medium sandpaper. Sand it just enough so it doesn't brightly reflect light, and the surface is clean and flat. Wipe up the dust.
Brush on a coat of flat primer-sealer, using a paintbrush. Make sure it's flat and smooth. Let it dry overnight.
Brush on a thin, smooth layer of finish paint, with the brush strokes all going in the same direction. Don't try to completely cover the primer, as you will add extra layers of paint later. Let the first layer dry to the touch.
Add another layer of finish paint, again keeping it thin and smooth. Let it dry. Add a third layer if necessary for complete coverage. Let it dry overnight.
Brush on polyurethane in the same manner as you did the paint. Let it set for a day.
Sand the dried polyurethane lightly by hand, using fine sandpaper, so the surface is slightly de-glossed, which will allow the next layer of polyurethane to adhere. Wipe up the dust.
Brush on a second coat of polyurethane. Repeat the process, letting it dry, dulling the shine with fine sandpaper, and then brushing on a third layer. Let the third layer set for several days before using the area.