http://www.gunshopfinder.com/articles/winchester59.html
The link above will take you to an article about the Win Model 59. Seems to me that it'd be a good little shotgun, and if it's in the shape you say it is, I wouldn't hesitate to soot it or hunt with it. I probably wouldn't shoot any steel shot through it, but as a super-light upland bird and small game gun, it looks like it would be perfect. As for the barrel, it's a thinner than normal steel barrel liner wrapped in fiberglass and bonded with epoxy. This makes it very light, and as long as there are no defects in the fiberglass, it should be plenty strong. There are modern rifle barrels that are made the same way that easily withstand 60Kpsi+. Shotguns typically operate in the 10-15Kpsi range, with about 20Kpsi being the upper limit on the heaviest magnum loads.
Mike