City transportation
It sounds very strange, but Fujairah's transportation is represented only by cab services. There are virtually no buses: if you see a bus on the street, it belongs either to a private company that delivers employees or to an educational institution.
So you either walk around the city (not always convenient though, the roads are wide and often dusty), rent a car, or your transportation in Fujairah is cabs. You can either use a specific service or simply 'hail' a yellow car with a green stripe on the side of the road.
Fare
The prices for this mode of transportation in Fujairah are low: a trip around the city will hardly cost more than 15 AED. For Fujairah's transportation, the advice to tourists is to choose municipal carriers because they have air-conditioning and meters. But many cab drivers don't speak English, so stock up on a map of the city so you can simply point to your destination in a difficult situation.