。
She walked past without noticing me.
Literal
She [topic-は] me [to-に] notice-not [without-ずに] walked-past.
気がつく ('to notice, to become aware') is a body-and-mind idiom — literally '(one's) spirit attaches' — for becoming conscious of something. The negative adverbial ~ずに ('without doing X') is a more literary alternative to ~ないで, derived from the classical negative ~ず. Common in written Japanese and in slightly elevated speech. 通り過ぎる ('to pass through and beyond') is itself a compound: 通る ('pass through') + 過ぎる ('exceed, pass'), giving the sense of going past without stopping.