In the movie The Wolverine, the minister of justice tells Wolverine that Yashida was obsessed with Wolverine and spent all of his money to stockpile adamantium in the north. But Yashida had not met with Wolverine since the dropping of atom bomb in Nagasaki. And in Nagasaki, Wolverine was without adamantium claws. Then how did Yashida come to know about Adamantium and its protective powers?
Largely assuming that The Wolverine is in the same continuity/timeline as X-Men 1-3
Despite wanting to generally keep a low profile, Wolverine was not a complete secret to the public. He popped his adamantium-covered claws in front of humans in multiple instances - including during a very public battle in San Francisco.
Regardless of whether or not The Wolverine is in the same continuity/timeline as X-Men 1-3
Yukio tells Logan that she has been searching for him for over a year. This tells us that they've been digging into his past and generally tracking him. The fact that his claws had been covered in adamantium would likely come up during the search.
Yashida is the head of a very powerful corporation and likely has deep connections with access to restricted information. If they're digging for information on him, or tracking him, a list of his current abilities/traits is more than likely going to surface.
chickenWolverine or the egg possibility
Tying back to the fact that Yashida is well connected and probably has access to information otherwise inaccessible to the public, there is also a possibility that Yashida learned of Wolverine's adamantium during his search for adamantium. Researching adamantium and its traits would inevitably lead him (or his scientists/researchers) to learn about subjects who had had their skeletons laced with it.