"this appears to be the 318 bore so i need 8x57JR it looks like JRS is for the new 323 bore diameter correct?" Yes.
My understanding of the "S" designation is that it specifies when the German military went from the old .318" dia. heavy round nose bullet to the newer high speed, lighter and slightly larger diameter "Spitzer" bullet. You can read up on that history elsewhere.
I had an old pre war Sauer drilling set up with an 8x57JR (.318") barrel under the twin 16 ga tubes. The really interesting thing was that it also had a plunger in the extractor, and would fire and extract rimless rounds as well. I think that one is for sale right now at Prophet river.
Now to make it even more complicated, give some background info and perhaps to offer an easy solution... both rimmed and rimless versions of the exact same cartridge case are made in Germany to European CIP specifications. The only difference is the designation "R" or not, and whether the rim is flush with the case wall ("rimless") or the rim protrudes ("rimmed").
So there are actually four version of this cartridge - 8x57J, 8x57JS, 8x57JR and 8x57JRS. confused yet?
Modern 8x57JS and 8x57JRS ( rimmed) are both loaded to higher pressure than the older and smaller diameter bullet 8x57J and 8x57JR.
North American manufacturers have chosen to ignore the difference between rimless 8x57JS and 8x57J ammunition. In North America all 8x57 ammo is loaded to SAAMI specifications and is called "8mm Mauser" This ammo is loaded with an in-between size .320" bullet and low pressure /low velocity 170 gr. RN loads, so it can be safely shot in rifles with either bore size. Performance suffers, it is not quite as accurate or as fast, but it is safe. Some European ammo imported by companies like PPU are also designated 8mm Mauser and loaded to low pressure.
So, If you are lucky and have the extra little plunger extractor in your drilling, you can load and fire 8mm mauser ammo in your old rifle, and i predict it will be safe to shoot. No prediction on the accuracy but you can buy 8mm mauser ammo quite readily almost anywhere.