Yes. Bourgain has a sum-product estimate for residues of a general modulus (although, I believe the case of a composite modulus with few prime factors that covers your question was worked out prior to this) See:
J. Bourgain, Sum-product theorems and exponential sum bounds in residue classes for general modulus. C. R. Math. Acad. Sci. Paris 344 (2007), no. 6, 349–352