Dust it regularly with a soft, dry brush, a lint roller, or a vacuum with the upholstery attachment on its lowest setting, working from top to bottom and following the direction of the fabric. Regular dusting is what keeps a lampshade looking new, since dust that settles into the weave is far harder to lift later.
For a mark, blot gently with a white cloth barely dampened with water, then let it dry away from heat. Never soak the fabric, rub hard, or use household cleaners or solvents, which can lift the printed color and loosen the lining. If a mark will not come out, send us a photo before trying anything further and we will advise.