There's “Yu” which is a short name for both Japanese and Korean.

“Yū” in Japanese long name:

• “Yūta” (ゆうた)
• “Yūji” (ゆうじ)
• “Yūsuke” (ゆうすけ)
• “Yūki” (ゆうき)

“Yoo” in Korean long name:

• “Yoo-seok” (유석)
• “Yoo-geon” (유건)
• “Yoo-bin” (유빈)
• “Yoo-jin” (유진)

Another idea could be aligning names from Japanese Kanji to Korean Hanja.

[* Kanji ≈ Hanja since they both mean “Hàn (Chinese ) Characters”]

A good example would be

Japanese:「しょうた」/shō ta/: “Shota”

Korean: “승태” /sŭng taé/: “Seung-tae”