Before you do anything, double check the diameter of the bar clamp on the Cinelli. The Nitto will be 26.0. But for many years, Cinelli stems were not interchangeable with any other stem (or bar) because Cinelli, and only Cinelli, used a 26-point-something mm diameter for the stems and bars where everyone else used 26.0 or 25.4. The difference between the Cinelli standard and the 3ttt/everybody-else standard was/is enough to make mixing a Cinelli bar with someone else's stem (or vice versa) somewhere between "hard to do and a bad idea" and "impossible." A friend did manage to get a Cinelli bar into a 26.0 3ttt stem 40 or more years ago - it made for more than a few quizzical looks from our other cycling buddies. To this day, if you ask him how it did it all he will say is "it wasn't easy."

Cinelli changed to 26.0 at some point, but I don't remember when (in the 1980s, I think, but I am not at all sure - someone else here will know) and I don't know if the 1A ever came in a 26.0 clamp diameter. I am all but certain that the 1A was first made with the 26+mm clamp.

I'm not saying you shouldn't go the Cinelli route - they look great and, assuming no cracks or other damage, are pretty darn good. Just save yourself some potential grief by doing some homework before you decide to go the Cinelli route.