I’d say fly, and if you can, go with Oman Air rather than one of the budget carriers. It’s only about an hour, but when I took flydubai I had a long wait at the airport, while Oman Air felt more seamless and gave a bit of that “luxury” touch.
For getting there, I used Emirates miles from NYC and transferred Amex points into business — Emirates is honestly one of the best airlines for cashing in points, so that worked out really well.
I’m not sure I ever made it all the way down to Salalah, though we did take a seven-hour drive. What I’d really recommend is renting a 4x4 and driving across the country at least once. Oman is extraordinary by road — the landscapes, the people, the whole rhythm of it. Flying is efficient, but you’d be missing somethin