。
Write down his phone number before you forget.
Literal
Forget-not while [in-に] his phone-number [object-を] note-down-in-advance [imperative-なさい].
~ないうちに means 'while still not [X] / before [X] happens' — a time-boundary expression. Combined with ~ておく (preparation) + なさい (soft imperative), it produces a natural parental directive. Contrasts with the ~ないように pattern in nearby sentences: うちに focuses on 'while memory is fresh' and ように on 'so that you don't forget.'