Will a .358 Norma Magnum reamer clean up a .350 Remington Magnum chamber?
No problem with the magazine length. The action is a Weatherby Mk V.

Per Ammoguide "yes"

Sometimes yes.Sometimes no. The shoulder on the 350 Remington Magnum is about .006 inch larger in diameter than the 358 Norma. The Norma case is 0.181 inch longer and the body taper is about .0025 inch per side per inch so some times it cleans up and some times it doesn't. It depends largely on how much over spec the parent chamber is.

If it doesn't, it will leave a slight ring about .200 inch down from the shoulder. If it does you will have to pick up the threads and set it back .200 inch to clean it up.

Shoulder diameter on the 350 Rem is .495" approx. 1.700 inches forward of the case head. The 358 Normas shoulder is only .490 " approx. 2.084" forward of the case head. So like Speerchucker said sometimes it may clean up the the 350 chamber and sometimes you may have a ring in the chamber. You may have to set the barrel back one turn to clean up the old chamber.

Easiest way might be to do a cerocast of your chamber so you can then do an accurate measure.