Since January 2018 the RAK Shuttle Bus connects Ras al Khaimah (or to be more precise: the hotels in RAK) with the Dubai Airport. There are three buses that service the route several times a day. The one way ticket costs AED 20.
As for now the buses from DXB serve all hotels as needed, while for the buses to DXB one bus serves the hotels in the city, one serves Mina al Arab and Al Hamra and one Marjan Island, so that the buses reach DXB fairly quickly.
Note: The schedule might change, as there are plans to add more connections to the schedule.
See www.rakshuttle.com for the schedule, further information and for booking your ticket.