You can buy the whole axle kit, which includes the cones and washers. A quick search in google for "axle and cone set" yielded several results from several reliable vendors. What I saw was in the $10-15 range. You'll need to sort through the specs to get the one you need.
As far as cone size...you may (or may not) run into compatibility with regard to the cone curvature and the current bearings. When I looked at one of the kits, it did not specify the size of bearing to be used...or at least I didn't find it.
Some of the kits appeared to come with the bearings, but then you may run into compatibility problems from the other direction, meaning the cup is the wrong size for the bearing.
I would say try to find a kit that includes bearings AND tells you what size they are. Measure the current bearings with a pair of calipers, and try to get a match. I expect with enough digging and sorting, you'll find what you need.
If you can't find a perfect match, and you are guessing about bearing size, I would use the bearing size that you currently have. That way you know they match the cups. I would suggest replace them with new ones of identical size. The only thing I thing you'll be risking is premature wear on the cones, but you might consider getting a second opinion on this. I'm not an expert.
Lastly, get in there an get a good look at the cups. If there is pitting or unevenness this is both good and bad. Bad because it can't be fixed, but good because it means you shouldn't spend too much time or money on it. The hub will never spin as buttery smooth as we want it to. When you tighten the cones, they'll either be too loose and there will be play, or they'll be too tight, and the hub will be crunchy.
If the bike is for occasional use, or just to see if you'll like riding it, go for it. If it is something that you hope will be a workhorse, and have longevity, replacing the hub or the wheel may be a better option.