。
Last year, she encountered Japanese culture for the first time.
Literal
She [topic-は] last-year for-the-first-time Japanese-culture [contact-に] touched.
触れる ('to touch') takes a metaphorical leap when paired with abstract things like 文化 ('culture'), 問題 ('a topic'), or 法律 ('the law'): it shifts from physical contact to mental contact — encountering, brushing up against, getting an initial taste of. The particle に marks the target of that contact (the surface being touched, in the abstract). 初めて always emphasizes the first instance of an experience and pairs naturally with verbs of perception, learning, or contact: 初めて聞いた, 初めて見た, 初めて行った.