There are lots of laundries - the cost to launder and get a shirt pressed is probably around 4 or 5 Riyals, 2 Riyals if you just want it pressed.

Don't know about dry cleaning but I would be picky about where I left my clothes. Magic Laundry beside the Giant Store in Hyatt Plaza seems popular, as does the laundry/dry cleaner outside Carrefour at City Center.