Beamforming design is an important technique for enhancing the performance of integrated sensing and communication (ISAC) systems. However, related research based on the hybrid analog-digital (HAD) architecture is still limited. In this paper, we investigate the partially-connected hybrid beamforming design for multi-user ISAC systems. Instead of the commonly used beampattern related metric, the Cramér-Rao bound (CRB) is employed as the sensing performance metric for direction of arrival (DOA) estimation. We aim to minimize the CRB while satisfying the signal-to-interference-plus-noise ratio (SINR) constraints for individual communication users by jointly optimizing the digital and analog beamformers. Subsequently, we propose an alternating optimization based framework, which is significantly different from the conventional methods based on the approximation of the optimal fully-digital beamformer with a hybrid one. We also consider an alternative formulation of optimizing the SINR of radar echo signals. Based on optimal receive beamformer design, we transform the SINR based joint transmitter and receiver optimization problem to a series of problems sharing a similar form with the CRB based transmitter optimization problem, which can be efficiently solved via the proposed algorithm. Simulation results show that the proposed designs provide significant performance gains in DOA estimation over the existing beampattern approximation based design.