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To get from Dubai to Muscat, you have any number of choices:
Flights (there are more than 16 flights a day between Dubai and Muscat).
You can drive there by car (you'll need special insurance to cross the border, and if renting, you'll need extra coverage); it takes between 5:30 and 6 hours.
The Oman National Transport Company runs a bus service as well; the buses run 7:30, 3:30 and 4:30 from Dubai. There is also Comfort Line that you can contact who also run bus services. You can find their offices in Deira.