The Long Action and Magnum receivers are the same dimension, but adapting a bolt is no simple task.
I don't expect your project would be too cost effective, Adam.
You would be better off buying a new bolt than machining a .30-06 .473 head diameter bolt to .534 for a RUM ctg. Remington bolts are hardened, and you would probably need to have the bolt machined for a Sako extractor. Machine shop work is not cheap if it is done right. Many gunsmiths won't do these jobs. Remington will not sell you a new bolt without considerable finagling.
For about $200 you can probably get a new bolt on Ebay.
Do you have a barrel? Be leery of buying a used barrel because after only a 4-500 rounds in a magnum chambering, there will be appreciable wear and the barrel is about halfway gone.
There are shops that manufacture replacement bolts, but they sell a bolt body only and such a bolt would need to have a handle added, and a complete firing pin unit, plus some custom fitting.
Good luck with your project. You'd probably be better off finding a used Ultra Mag in a gunshop. Lotsa guys trade them in after they experience the recoil. If you are really 16/17 yrs old you have time to save your money. Better to spend an extra hundred or two up front.
Sometimes gunsmiths botch jobs. Sako extractor-conversion jobs are not as safe as the original Remington design. Better to find a used 338 or 300 RUM, and shoot the sporter barrel out as you get to know the cartridge. You can likely find a good stainless RUM with a scope for $500 or less. Good luck with your project.