。
She walked right past me without noticing.
Literal
She [topic-は] not-noticing me [possessive-の] front [path-を] passed-by.
気づかずに combines 気づく ('to notice') with the negative continuative ~ずに ('without doing'), giving 'without noticing.' This ~ずに form is the more written-style cousin of ~ないで, both attaching after the verb's negative stem. 通り過ぎる is a compound verb: 通る ('pass') + 過ぎる ('exceed, go past'); the ~過ぎる auxiliary adds the sense of overshooting or going beyond. The を on 私の前 marks the path of motion, the same use that appears with 道を歩く ('walk along the road') or 橋を渡る ('cross the bridge').