The other guys have it right on trying to ride with the rifle in your Eberlestock
pack. It will be uncomfortable, snag on trees and the such and if you are going with an outfitter, they may not allow it.
The Trailmax rifle scabbard Bucksnort mentioned will do the job for you. I have one and have used it on several horseback hunting trips. It is a cordura, full containment scabbard and will hold your rifle securely. I have used mine with my 338khan which has a 30" barrel + a brake, A5 stock and a 8-32X56 NF scope, all weighing in at about 17lbs. That rifle fits, although not with a lot of room to spare. The scabbard also makes a pretty good soft sided rifle case for just carrying the rifle even when horses are not going to be involved. A problem you will have with a large heavy rifle in a scabbard is the saddle will keep slipping toward the side with the rifle because of the extra weight, and it will be uncomfortable under your leg if you mount the scabbard horizontal. No real fix for the saddle slipping, you will just have to stand on the opposing stirrup to rotate the saddle back around every little bit, and may have to dismount and let off the cinch and reposition the saddle every now and then. Large heavy rifles and horsepacking create some challenges, but it can be done.
As for carrying the rifle I prefer the Kifaru "gun bearer" system. It can be put on about any pack and gives you much faster access to your rifle than the Eberlestock. I know fast access is not normally an issue with long range hunting but the real advantage for me is for those "take a breather" moments going up a mountain. It is real easy to get the rifle in and out of the gun bearer so I can get that 17lbs of weight off my hips for a couple of min. If you try it, with the large NF scope you will need to carry the rifle "backward" as to their instructions. In other words the scope will face forward. Large scopes such as the NF do not fit well under your arm where they normally would go. It will take some experimenting to get everything positioned right, but for me anyway it is comfortable way to carry my big rifles.