Oh, I basically did this in July, from mid-afternoon until late evening when my flight out happened.
Assuming you don't want to just grab a cab and go down to the waterfront (it's nice and there are cafes and the market), the airport doesn't have much. At all.
You have a choice of two Costa Coffees, and a small restaurant place with some Indian-like food (much Omani cuisine is inspired by Indian food). It's nothing fancy. That's it. The rest of the shops are cellphone sim card stalls, and care-hire, basically.
Going through check-in, you get to go upstairs, and this is bigger - but now you're going to have Wendy's, Subway, and that style - plus one small restaurant that serves shish kabob and wraps. And souvenir shops and exchange places. It was pretty limited. They also claimed to have free wifi, but it wasn't working when I was there. All day.
My recommendation - landside, get into that restaurant (it's opposite the small Costa), and settle in at one of the booths. Each booth has a powerpoint, and you can just chill and watch movies on your device or listen to music, while people watching a bit and occasionally buying a drink.
Otherwise though, no, I did a lot of laps walking, but there's really not much else to see. If you can, you're far better off adventuring outside, even if for a short time. Town is pretty close - go see the new mosque at night, it's quite a sight!