As Sjiveru says, it's an adverb modifying a verb phrase. Saying that い-adjectives change to く form before a verb is an oversimplification; く form is the adverbial form.
This a tricky sentence because we would use an adjective in English, so let's look at another example.
I ran quickly to the store.
As you can see, "quickly" is an adverb modifying "ran." Now, in Japanese:
The word order here is perhaps a bit awkward, but you can see that 速く isn't modifying 店, it's modifying 走った. If you changed it to:
It wouldn't make any sense. What's a fast store?
Now, to the original headline:
Fujii-san is the youngest Shōgi player to attain 7th level. Or in headline-speak "Shōgi's Fujii Souta-san youngest to attain 7th level"
If you changed 若く to an adjective, it would mean that Fujii-san is "becoming the youngest 7th-level." This is awkward if not ungrammatical, because unlike, say, "black-belt," "7th-level" is somewhat unnatural as a metonymy for "7th-level player".
Note: I know nothing about Shōgi, so I'm not sure if "7th-level" is a natural translation or if we'd say "7-dan" or something else in English!