There are tons of mosques in Oman. Most of them are pretty new. You'll even find them at gas stations sometimes. That said, at the most popular ones, they either limit tourist visits to morning hours—like 8 to 11 AM—or don't let them in at all.

This was the first time I'd seen visitors allowed not just to check out the interior, but actually set up a separate tea area where they serve free tea and coffee and answer any questions you have about Islam. If you're really interested, they even offer literature. Some of it's in Russian too.

Hotels usually have a sign pointing toward the Qibla so guests know which direction to pray. And there are radio stations that just play Quran reciters nonstop (though there are music stations too, of course).

By the way, people in Oman aren't Shia or Sunni (if you want to know the difference, redroom has a pretty good explanation), they're Ibadis. The main differences with Ibadis are pretty subtle. Like, most people believe you can go to paradise from hell once you've atoned for your sins, but Ibadis think once you're there, that's it. There's no coming back.