My first WoW/Lua project was updating an addon whose author had stopped maintaining it. I've always found it easier to learn by looking at something that already works, than to try to start from scratch with an unfamiliar subject, and "Hello World" type tutorials just make me want to quit the subject. The same applies to non-programming things as well... I couldn't build a car if my life depended on it, but I can fix the one I already have.
I'd suggest just running the addon in-game. If it works, just keep using it. If something doesn't work, there's probably
an error message, which will tell you exactly what and where the error is. If you need help figuring out how to fix it, post the error message here.
As for whether or not you're allowed to post a "fan update", that depends on the addon and how it's licensed. If there's no license listed in the addon (GPL, MIT license, etc.) then it's "all rights reserved" and you're not allowed to publish updates, even if the original author has disappeared. You can fix your own copy for your own use (maybe not legally, but in reality) but that's it. In that case, if you want to share your work with the community, you'd need to write a new addon from scratch.