I've had an old Trek 1000 sitting in my basement for years. I've got a rear rack on it and used to carry stuff in panniers on this bike years ago, though I don't think I ever did anything more than a long weekend trip on it. But if I HAD to, or WANTED to, I wouldn't hesitate to head off for a month on this bike. With my own Trek 1000, my biggest concern would be the narrow tires and the gearing-the lack of a triple chainring. Both of those can be modified to some degree, however. Anyway, I don't see what the big deal is. What are you specifically concerned about?
I once flew off to Germany for a month-long tour with a German friend. The airline lost my bike and it was never found. My friend was hit by a car the year before, and although he wasn't hurt, his bike was damaged. It was an old Raleigh, and he had bought a new bike after the accident. We got his old Raleigh out. I had to get a new rear wheel, and there was some visible damage to the frame, but the bike was perfectly rideable with a new wheel. Oh, it was far too big for me, as my friend is much taller than me. My leg extension was OK with the seat as far down as it would go. (at my present age, my biggest concern would be the incorrect frame size, but I was young, stupid, and my knees could take the too big frame then). Anyway, we went off for a month of superb riding in France--Auvergne, Dordogne, Pyrenees, a bit of Spain. One of the best trips I ever took, and no shortage of hills and mountains. The bike was a half-broken piece of crap, but it was still a bike and it worked. Your Trek 1000 sounds like it is in better working condition than that old Raleigh was.