Not sure I'm following you -- yes, the Queen knighted him. How does the multicultural part come in? (Women receive the same award but it's called a Damehood and their prefix is Dame.)
If Kazuo Ishiguro is knighted, should he be addressed as Sir Kazuo or Sir Ishiguro? The point being that people of Japanese descent might place their first names after their last names.
This question doesn't seem very relevant to knighthoods. Whatever would be appropriate to call someone as their given name, rather than family name, would go after "Sir". It's not very complicated.
Sir Kazuo. With some exceptions - Japanese people "normalize" their names to English name order of Given Name, Family Name when writing or speaking in English. Some choose not to, usually out of national pride, but it is most common to normalize the name to the target language.
A sidebar issue: all these rules apply to knighthoods from the UK crown only. And under these rules non-Brits may be knighted but do not use the title 'sir'. Thus Bill Gates KBE is not known as Sir Bill.
(edit: Today I learned that Kazuo Ishiguro is British. The statement below is completely wrong. I've managed to get his nationality wrong while enjoying his books for 20 years!)
So in the specific case of Kazuo Ishiguro, the sir does not apply. For Brits with family-first names, see the other answers.
That's a language issue, not a culture issue. Normally, one uses the name ordering appropriate to the language one is speaking at the moment. Therefore, the question is equivalent to "What is the Japanese translation of 'Sir'"