Having shot several different .338mags, I can lend a little insight.
Recoil is dependant on the bullet weight, velocity of that bullet, and the rifle to include stock design, composition, and recoil pad.
The first .338mag I shot was a Remington M700. Recoil wasn't that bad. Significant, but manageable/tolerable. Likewise the Winchester M70.
Then I shot a Browning Stainless Stalker w/26"bbl and synthetic stock. The owner wanted to chronograph some handloads with 250gr Nosler Partitions over a book max load of IMR4350. I shot them through the chrono from the bench as I fear someone blasting my chronograph due to flinching. Recoil was very mild considering the cartridge. I wasn't too impressed with the velocity as it was 2,650fps. He was pleased as that was what the book said it should get. Not a bad load however.
My personal .338's are a custom barreled Interarms MkX Mauser in .338/06. I get 2,600fps with 58.0gr of IMR4350 and near one-MOA groups at 100yds. The other is a Marlin .338MX. It with certain loads will shoot sub-moa for 3-shots. It gets 2,500fps with a 200gr bullet, but is a deer killer supreme.
The Mauser was the worst kicking .338 I'd ever shot. When it was a .30/06, it was the worst kicking .30/06 I ever shot. I used to have a bunch of Norma 220gr factory loads that I shot up for the brass in the late '70's. My aquaintainces would talk about their hard kicking rifles, usually a 7mmRemMag. I'd challenge them to shoot my rifle. None would shoot the '06 more than once! They'd say "you win!" The .338/06 barrel kicked even worse, even after a recoil pad was added. The design of the original walnut stock somehow made the recoil come right at you. No absorbtion and a hard butt plate, I suppose.
However after putting the action in a Hogue stock, the rifles character was totally changed. It's now more pleasant than most of my 7mm or larger caliber rifles. Recoil is there, but is the "push" rather than "whack" it used to be.
I really like the .338 caliber. The bullets tend to have excellent ballistic coefficients and the 250 and heavier bullets have high sectional densities. However, with a bullet such as the 210gr Nosler Partition they can be very destructive on smaller big-game such as our ~100lb deer. The worst I ever shot up a deer was with my .338/06 and a Noser 210 Partition over 60.0gr of H414 at a chrono'd 2,750fps (duplicates Weatherby factory load). The shoulder/spine shot almost blew the deer in half. I lost both shoulders, nearly half the back-straps. It was definitely more destructive than any load I've shot a deer with from my .300RemUltMag.
The .338's are a lot of gun. I believe you'll enjoy one, especially if you reload. If you don't reload, you won't shoot much, not because of recoil from the rifle, but the RECOIL from the PRICE OF THE FACTORY AMMO!.....
OUCH! AT THE CHECKOUT COUNTER !!!!!