How would one go about updating a 1st gen Apple Watch running watchOS 1.0?

You're going to find out it's almost impossible to update a series 1 watch. In theory, they should work with just about any iPhone, but an 8 + year old watch, with such outdated internals and extremely limited memory is going to be a difficult thing to update at best.

If it can be updated, it would be advisable to schedule an appointment at your Apple Store Genius Bar and see if they can possibly update the watch. Even Apple may not be able to do it.

As I mentioned, just about any iPhone will theoretically work, including the newest generation iPhone's. It's the watch that's the problem. So your in theory iPhone 5C would work with a Series 0 watch. Though I don't think it would be remotely possible to purchase a brand new in a sealed box iPhone 5C, which was released in 2013 (ten years ago) and discontinued in 2015, everywhere except India, where it was available until 2016. Nor should you even consider buying such an old, no longer supported completely out of date iPhone. So many features simply won't even work on such an outdated phone. So many.

You can spend your money on anything you want. But I think you'll find these vintage, no longer supported phones, which won't even likely connect to the internet won't provide you much benefit. You won't be able to download almost ANY modern Apps. No Netflix. No Youtube. So many things you can use a modern iPhone to do, these vintage devices will be useless. Including for making a phone call. 3G service, which these older phones used has for the most part been taken down throughout most of the world. If you can get your hands on a never opened original iPhone, good for you. But a used one, is pretty much all you could waste your money on. And waste your money, you'd surely do.