This paper considers robust codebook-based downlink beamforming (i.e., single-layer precoding), where the beamformer of each user is chosen from a fixed beamformer codebook defined, e.g., in LTE and LTE-A. Admission control and power allocation are embedded in the precoding vector selection procedure. The objective is to maximize the system utility, defined as the revenue gained from admitting users minus the cost for the transmitted power of the base station. We adopt the quality-of-service constrained approach and the robustness against channel covariance estimation errors is realized with worst-case design. The robust codebook-based beamforming problem, which is a bi-level mixed integer program, is converted into a more tractable mixed integer second-order cone program. Techniques are proposed to customize the convex continuous relaxation based branch-and-cut algorithm to compute the optimal solutions. A low-complexity inflation procedure is also developed to compute the near-optimal solutions for practical applications. Numerical examples show that the gap between the average number of admitted users achieved by the fast inflation procedure and that of the optimal solutions is less than 11.6% for all considered simulation settings. Further, the inflation procedure yields optimal solutions in 88% of the Monte Carlo runs under specific parameter settings.