The long distance bus station is not all that far from the Sheraton... 10 min walk? just down the road perpendicular to it. I would imagine there are buses to the airport from there but I only used that bus station for trips to Nizwa, Ruataq and Salalah. I can't recall any particular Persian restaurant. It's been a while since I used the Muscat airport but all taxis were a "negotiate the fare" within town but I think the airport ones had meters. Sorry, can't recall.

You should still grab a taxi over the hill to Mutrah and walk along the corniche to the souk. A couple hours would be enough time to get a look.