It's about an hour to the roundabout where you enter RAK (and would turn right to go towards RAK City) if you stay on Emirates Road. If you go on the E611/bypass road it's about 25 mins from the end of the E611 to that roundabout.
From that roundabout it's 52km to the border crossing and the route is not fast. It took me almost an hour to go not-quite-as-far as the border.
This is if you're heading to Khasab.
From AR we can be in RAK in an hour if no traffic on the Emirates Rd. You can add ages if you're trying to get through Sharjah during busy times. You could try the by-pass road (611). From there it is probably 15 mins to the border but it's not clearly signposted. You then have to do the border crossing and that depends how busy it is. Then probably 40 mins to Khasab.
HTH