Hotels do this as a matter of course as the visa costs them money. Who are you flying with? Emirates will sort your visa if you ask them to, not sure about others but airlines are likely your best first option. As mentioned above, there are also agencies who you can pay and they will sponsor you (you don't need to book anything else through them), though they tend to be quite expensive and technically this isn't legal.

The visa is for the whole UAE regardless so you can travel to any other Emirate once you get it.