Micromesh was originally developed to polish scratches on the plastic cockpit windows of aircraft. As such, it's best used on plastic - which in the turning world means objects turned from acrylic, polyester or other plastic materials, or things turned in wood and finished with a hard film finish such as CA. I've used it on wooden items finished with both polyurethane and alkyd varnishes, and I don't see any reason why it wouldn't work on things finished with lacquer. I would probably stay away from things finished with shellac because it is best used with water as a lubricant, and water and shellac don't play well together.
As others have said, work through the grits with conventional abrasives up to about 600, then switch to micromesh. Most people use one of those plastic snap-lid disposable food containers as a water reservoir, dipping the micromesh into the water and then buffing the workpiece with wet micromesh. It's a good idea to use a bit of paper towel or toilet paper to wipe away the slurry between grits whilc working your way through the colors up to 12000 grit.
But to be frank, I suspect that using one of the abrasive pastes (eg, Yorkshire Grit) followed by wax might produce essentially the same result. and be both faster and less messy.