Here's a pic of my 1980 CX500 "Custom" after I pulled it out of a garage and did a lot of work on it. These are great old bikes, but I fell about laughing when I saw some references to "Fast" in previous posts. These things weigh as much as a CB750 with only around 48 horsepower! The only way you'll get one to do the ton is to drop it down a mineshaft, ha ha!
But they are a good honest old thing, not bulletproof, but if the oil and coolant (don't use the acid based stuff or it'll eat all the solder in your copper radiator and it'll fall apart, use the cheapest green stuff and it'll be fine, check with one of the guys on the CX/GL500 forums, they know more about these old boat anchors than anyone here.
http://cx500forum.com/ The carbs are crap and an absolute baastard to clean, I mean, the worst crabs to clean in the world, bar none. If you want to make it go "fast" (a relative term for a CX500) dump them and contact Murray Feldman in North Carolina and buy a set of his 34mm Mikuni's like the ones on my bike. That'll fix one end, but to make it breathe properly you need to also bin the original crossover pipe under the engine and the OEM mufflers, the crossover piece is a muffler in itself, and restricts the crap out of that little engine, it really needs it.
Apart from that, the early models have expensive ignition problems, (requiring the engine to be pulled out to fix) but surprisingly they handle quite well, and with Murray's carbs and better exhaust they are great fun to ride, and I loved mine, but I needed the cash to buy my BMW so I sold my little CX500 Custom and was surprised at how much I was able to sell it for.
I've got another one here, it's a 1979 Shadow like yours, that my son and I am building into a "Bobber", it was in really rough shape, so I don't feel guilty about chopping off the horrible rear frame section under the seat and making it look a lot more attractive. (another word not usually associated with "CX500", ha ha!) Have fun with it mate, that one looks so clean you cold just keep it original and once you get sick of being dragged off by Hyundai's, sell it for more money than you paid for it! Cheers, Terry.