Sure you can.

Air Arabia flies quite cheaply from Sharjah, otherwise Abu Dhabi seems to have cheaper flights to MCT than Dubai. Look out for fifth-freedom "tag flights" from European carriers e.g. BA (AUH) and Swiss (DXB), can be very attractively priced (although they get in to MCT late).

Otherwise you can drive: a more expensive choice, but it gives you much more freedom and can be part of the holiday. You need to watch out for the insurance fees covering you for Oman. Driving means you get to see Al Ain, the sort of spiritual homeland of the Abu Dhabi emirate, and you can then route yourself through the mountains and Nizwa to get to Muscat.