Midnight to 1pm is a tough call. If it were me, and knowing that you have time to explore properly on your return trip, I'd grab a room at one of the really cheapy hotels down on the Muttrah corniche and get my head down til 6:00, because there will be nothing constructive to do in the small hours. Get up with the sun rise and you'll be in exactly the right location to go for a wander in the souk and port area, and watch the corniche wake up. Maybe take a taxi over to the palace complex in Muscat proper, and then come back to Muttrah when the shops in the souk open up between 9:00 and 10:00, and then head back to the airport at around 11:00 for your next flight.
The Ramee Guestline is walking distance (less than 1km) from the Crowne Plaza and is excellent value for money, although it can be very noisy at night - it has a bar / club on the ground floor thats a very popular expat hangout, so ask for a room on a higher floor. I had friends here over Christmas who got rooms for 40 OR a night, and were really happy with that.