I have a 1982 Nishiki Sport steel frame road bike. I broke a spoke on the rear wheel. I would love some specific recommendations for the correct spoke to replace it with. 27" x 1 1/4" wheel, 32 spokes. The manual says that the spokes are "080 GA UCP Finish"
If you still have the old spoke and both/all its parts then that can be used for measuring the length.
If that's not possible, as per Daniel's comment, then remove the next spoke that goes to the same side of the hub as the broken spoke - this is important on the rear wheel because the two sides can vary by several millimetres.
If you're buying one spoke, get three and clearly label the spares. There's a non-zero chance that others will break having been stressed a bit more, and a ~40 year old spoke has probably been around a while. You don't need a whole box of 50, but they can come in lots of 18 which might work out a good deal.
Owning at least one spare spoke helps protect the ones on the bike :)
You'll also want to find the same:
overall shape - its probably an unbutted spoke, meaning its the same width all the way down the length. Some spokes can be described as double butted or even triple, depending on how they taper.
Ends - the thread at the nipple end is universal, so that's okay. The hub end is very likely to be a J bend with a mushroom head, this is 99% normal too.
Profile - its also likely to have normal round spokes, not fancy bladed or aero spokes.
Finish - spokes from the 80s could be shiny chrome or polished finish, or utility galvanised finish. Black spokes weren't very common. You might choose to match the finish.
Nipples - they can round off over time, faster with a slack-tolerance tool. Consider getting some new brass nipples with your new spokes. That said, I've often reused the old nipples without issue, other than the risk of rounding.
Spoke prep - Almost a paradox, spoke prep is an assembly lube and friction increasing agent at the same time. It helps ease the two together, and assists the nipple to not back out under use. Personally I've never owned the right stuff but some spokes come with threads pre-dipped in spoke prep. I've used copper-clay assembly paste and it works fine for me.
Regarding tools, check the Related questions on the right. There will be ~2 special ones you require, maybe 3, and some optional ones.
It is possible to calculate spoke length using a calculator like https://leonard.io/edd/ but there are a lot of measurements needed, which can be hard to do when the hub and rim are built into a wheel already.
To err on slightly too long is better than slightly too short, IMO. You can always file a spoke end flush with the nipple, you can't make them longer.
Aside - consider adding some of the new-style clip-on spoke reflectors. They can help balance your wheel when put in the right places, and more importantly, they work a lot better than the yellow rubbish plastic ones.