The '81 engine bolts in place of the '80, but you'll also need to swap in the CDI parts, the '81+ back wheel, two piece throttle cable, 81+ airbox, etc. That'll give you the much improved carb, combustion chamber and larger clutch. I'm not certain, but I think the NA50 had the 2 speed tranny in '81, while the NC50 got it in '82.
It's also possible to swap on an Urban Express engine, but it's even more work. With the extra work comes better results though.
I would suggest taking a look at the '82 Express to see if you'd rather just get one of those instead of trying to find an engine from one and going through all the work of putting it on your frame.
The NU50 bore kit SHOULD work on the NC50 engines as far as I know, but I could be wrong.