Take a look at this,
Gulfnews: Conmen bribe alcohol buyers in Ajman
I know its a bit out of date but it does say;
"An official at Ajman Police told Gulf News that non-Muslims can buy alcohol for personal use only from outlets in Umm Al Quwain, Ajman, Dubai and Abu Dhabi if they have a liquor licence.
He added that non-Muslims who have legally bought alcohol are allowed to transport it for personal use only in their vehicles through any emirate, although they must not be under the influence of alcohol behind the wheel."