I am assuming you are thinking of a Prime Rib Roast? If you want to go cheaper there are 'other' cuts (chuck, etc) that work ok but will not be as decadent in terms of flavor and tenderness (for a rarer doneness). Most of the time, I cook any other roast to the past well point as pot roast.
The rule(s) of thumb essentially if the cut is able to be enjoyed rare you should cook it more conservatively in terms of time, otherwise you go to the other end of the spectrum a.k.a. low-and-slow (low temps, long time). If you want to go with a cheaper cut (and believe me I don't shell out the dough for a Prime Rib cut other than Christmas) then I would suggest a long braised affair. At any rate it must be a prime cut worthy of roasting (Rib, Sirloin, Tip) if you want a shorter, hotter roasting.
If you go the traditional route, a lot is riding on a nice roast so I would seriously consider using a meat thermometer if you haven't done it before. There are loads of methods out there, but how I learned is you essentially 'blanket' the fat cap of the prime roast with salt and pepper. A thick layer of each. It goes into a low oven for the appropriate time according to the weight. (don't recall off the top of my head). The hardest part is that you have to trust that thermometer and remove it about 15 (I have seen it go this high in the rest, most texts cite 5 to 10 ) degrees before your cut reaches the proper temp. So if you want it rare in the center (the ends will be more done) which is about 120 F then you remove it when it hits 105 and let it rest for at least half hour. A rare roast can be 'fixed', but an overdone one cannot be salvaged. You can start the oven high, like 450 and then lower it to about 325 (this varies, sometimes I go 250 for a very long time. It all depends on how long you want to spend keeping an eye on it) when you put the roast in or just go 325 the whole time. Either way
make sure you allow the roast to come to room temp, which takes about 2 to 3 hours for an average roast.
If you want to go with a more economical cut, the way I always go about it is, you get yourself a large oven-proof dish (a
casserole works best, invest in one if you like these types of things...it is worth it) and over a medium high heat, brown your roast all sides in a little oil or clarified butter. Remove the roast and then add 2 large sliced onions, thick is nice. Carmelize them by sprinkling with a little salt (to draw out the moisture) and then keep stirring them until they get gooey (a sign that the sugars are beginning to carmelize) and brown. Now add about 2 cups of dry red (or white) wine, or stock. Stir it briefly and then place your roast back in. You can add mushrooms (whole or diced) at this point as well, as well as season with salt, pepper, bay, thyme, etc. If you need to, add more liquid (You want to have essentially enough liquid to go about 1/4th the way up the roast from the bottom of the pan). Cover it and put it into a 250 oven for about 5 hours. Now if you want the meat falling apart or sliceable you have to pay attention to this during the process and time it. About an hour before you are done add chunked carrot, potato, rutabega, celery, etc, etc. There are many variations, but just place them right on top of the roast, cover and cook. Don't forget to season with salt and pepper as you go. In the end you'll be left with a wonderful jus, you can thicken it with a roux or potato starch. Serve this with some nice crusty bread or over noodles with sour cream.