The NS is a good gun. Enjoy.
A couple minor things...
First, like all older shotguns, a check over by a decent smith is a must. Besides the possibility of being off face or loose, there's some possible issues with triggers/sears and safeties that have never been cleaned.
A ring test will tell is something's coming undone in the barrel assembly.
Remove the barrels and hang them off your left index finger or a looped string. Tap them with something non marring like a pencil. If they ring clear, you're good to go. A buzz or dull thud can indicate something's wrong.
Most Nitros I've run across have been choked on the tight side. You may want to open the choke in one or both barrels for your type of hunting or clay slaying. This can help a lot when in the uplands.
A couple things I would never do with or to an NS.....
Cut the barrels. Built when even entry level doubles were made with some attention to balance, the NS swings well and points like a finger for most of us.
Use slugs or steel shot. Not a good tool for those and I'd hate to see a bulge happen.
Install choke tubes. While I like these on most guns, double guns this old and well made deserve preservation. If you want a do all, get a pump.
I would,since factory stocks tend to shortness and plain wood, bite the bullet financially and have Wenig make new lumber to my personal specs, but if it hits well for you as is leave it alone.
Enjoy your great old gun. You are the custodian....