I don't think anyone knows enough about the period to answer that question. Our main written source is the Annals of Spring and Autumn (春秋), from Lu (not Yue https://en.wikipedia.org/wiki/Yue_(state)) which spends 18000 words on 242 years, in chronological order. For many of those years it simply says something like "螽" ("locusts"). It does not go into any detail on questions like which hobbies the kings of other countries spent their spare time on, and that's a difficult thing to infer from archaeological evidence, too. The written records of Yue were destroyed by order of 秦始皇 in the 焚書坑儒.

So we kind of have to guess. My guesses are not the most informed.

The sword (https://en.wikipedia.org/wiki/Sword_of_Goujian) is bronze, so it was probably cast (you can forge bronze but the cost/benefit ratio is terrible). You could imagine a king pouring the hot bronze into the mold—that would be much quicker than forging an iron sword—but you probably wouldn't want him to make a habit of it, because contaminants in the metals would expose him to arsenic vapor, though this sword in particular is almost arsenic-free.

Then all that's left is sharpening the blade, which any warrior has to be good at, and what is a king if not a warrior foremost? So it's plausible that a king might have put in most of the work embodied in the blade himself, with a grindstone, even if he didn't go around casting bronze regularly.

It wouldn't be unprecedented. Every now and then you get a monarch who just happens to like doing something with his hands. It's rare enough that it's usually specifically noted when it happens, but e.g. consider Peter the Great of Russia, who famously partook in building ships for his fleet as a carpenter (although in his case it seems to have been driven more by relentless perfectionism and dissatisfaction with quality of other people's work).

Honestly I wouldn't be surprised if he actually made it himself (though I agree the inscription is hardly proof). We have this view of Kings as fat, lazy bureaucratic blowhards because we're closest to the tailend of the history of monarchy. When this King was around there were a lot more regional leaders who fought and strength was admired the most, if a King was seen as weak he had just as much internal attempts that were way more physical than the political coups of today. You could not be a king or survive being a king without being a good fighter and knowing a bit about how weapons are made and used.

When you have some sort of divine right to rule, the most money, a palace filled with servants and responsibilities that can be neglected with no immediate consequences, well you have a lot of free time.

That said, while it's possible this King was really into swordsmithing, more likely he's just taking credit for the work or something gets lost in the translation. Like if Elon said he "built the Falcon 9". It's not explicitly true, he certainly wasn't machining parts or writing code for it, but he was involved enough that no one would really call it a lie either.

Even recent kings have this grand education from the best of the best if not from actual geniuses, examples abound like Alexander the great and Aristotle, and usually there’s other examples in wikipedia articles for most recent kings, even without tutoring here’s a list of books that could have been used to educate royalty in less ancient times than this case but of course maybe a book wont teach craftsmanship https://en.m.wikipedia.org/wiki/Mirrors_for_princes

The elite/aristocracy of these old ages was at the top for reasons that make some sort of sense even though it could be violent it would be perhaps not completely uncorrelated with ability, if a prince is born into it they’d sometimes get taught multiple languages, music, poetry, on top of the reasonable politics and warfare, strategy etc. As for craftsmanship that is a good question since it’s working with ones hands but with the examples here so far it would make sense if they could just get instructed by the best craftsmen as well to learn that too of course this is maybe too far backwards and another continent to extrapolate

Well, everyone needs a hobby :)

Here's another one, different king, similar inscription: https://en.wikipedia.org/wiki/Spear_of_Fuchai - seems likely it should be read more as "caused this to be made".

Well kings have always said they made/build anything they ordered. I would simply not believe Yue's claim. It's much like the things Wolfram "creates".

Julius Caesar made the dang calendar. People had a lot of free time.