You can dry-age any large cut of beef when it's whole. I wouldn't recommend trying to age cut steaks, as the shrinkage and trimming would ruin them. Here's the process I have used:
Wash your prime rib and pat it dry. Be sure to wear gloves to keep the bacteria from your hands off the meat.
Wrap the meat in a couple layers of cheesecloth, and place in the fridge on a rack inside a baking pan. The idea is to keep all sides exposed to the air, and have a catch pan below. Some recommend putting rock salt in the pan below, but I don't.
Make sure you put it in a fridge that will maintain a consistent 34-35 degrees, and have the door opened as little as possible. Set a small thermometer next to the pan to monitor the temp.
Change the cheesecloth every day for the first 3 days. After the second or third day, the cloth won't be bloody anymore. A dark red "crust" will form on the meat, and you'll begin to see the fat cap begin to dry out as well.
Keep it in at least 8 -9 days. Some people dry-age for 28 days, but I never could wait that long! Inspect/change the cheesecloth every 3 days.
After 8 days, the meat will be pretty dry on the surface. When it's time to prep for cooking, trim all the crust off with a sharp knife. You'll expose a deep red meat that looks fantastic!
It will cook as usual, but will be rich in flavor, and fork tender!