I've never transited through RUH but I have arrived quite a few times this year.
I don't believe there are any specific routes or lines/queues for transit pax. The only route from the arriving aircraft is to immigration, the lines for which can be horrific depending on what time you arrive. In fact, I can't picture in my head how you'd get from arrivals to departures airside. At least you don't have to change terminals or spend much time there.
I would speak to the inbound SV crew and ask their advice on what to do. Don't fret as it is obviously possible, although as
DXBPlat has said, RUH isn't the most civilised or welcoming of airports. No real reason to avoid this though if it would mean flying on someone else and paying more money. Enjoy the adventure
Your wife won't need to cover up airside, but I would advise her to leave the shorts and t-shirt at home. Remember the concept of hijab refers to "modesty".
Please post back with your experiences.
hth