。
She'll likely become Japan's first astronaut.
Literal
She [topic-は] Japan-first's astronaut [become-となる] [probably-だろう].
~となる is the formal/written variant of ~になる ('become') — preferred in news, formal pronouncements, and historical narration. ~だろう is the conjecture auxiliary ('probably, presumably'). 日本初 ('Japan's first') is a productive pattern: 世界初, 業界初.