Which phrase is correct English usage?

B is almost perfect, but change "new" to "newly" and add "year" after "fiscal."

I would edit it this way: "Japan's newly designed 10,000-yen note is expected to go into circulation around the first half of fiscal year 2024."

In all cases "newly-designed", not "new designed"

Definitely not C - not "get into circulation". Apart from that A or B is suitable.

In Britain I would not expect to see "fiscal" there, it would be unnecessary and possibly not fully understood by some people. I would possibly expect: "....enter circulation during the first half of 2024". That date is so far in the future, and the notion of "half" is so indefinite in connection with the months of a year, that it would not matter whether it was the calendar year (starting in January) or the tax year, starting in April, which was intended.

Japan designed a new 10,000 yen note that is expected to circulate around the first semester of fiscal 2024.