I hurried to the airport so as not to miss the plane.
Literal
Airplane [onto-に] miss-not [so as not to-ように] I [topic-は] airport [to-へ] hurried.
~ないように expresses a negative purpose: 'so as not to X.' 乗り遅れない is the negative of the compound verb 乗り遅れる ('to miss a ride'). The word order places the purpose clause before the main clause, which is natural in Japanese.