I have been to Oman 3 times via road. It will take about 14/15 hours to reach Muscat. You will definately require Saudi visa (as they do not give visa on arrival) which you can get applying through any of the agents. below might help you for your route:

1 - Qatar-Saudi (about 50mins)add another 45mins to 1hr for immigrations & customs
2 - Saudi-UAE (about 80 mins) add another 45mins to 1hr for immigrations & customs
3 - UAE - Oman (5-6 hrs) add another 1hr-1-1/2hrs for immigrations & customs
4 - Border to Muscat (3-4hours)

Have a nice journey and all the best. last but not least avoid driving in night on highways.