Parameters of the honeycomb core, such as cell size and foil thickness, as well as the material of the core, influence the ballistic performance of honeycomb-core sandwich panels (HCSPs) in the case of hypervelocity impact (HVI) by orbital debris. Two predictive models that account for this influence have been developed in this study: a dedicated ballistic limit equation (BLE) and an artificial neural network (ANN) trained to predict the outcomes of HVI on HCSP. BLE fitting and ANN training was conducted using a database composed of entries resulting from physical and numerical experiments. The new ballistic limit equation was based on the Whipple shield BLE, in which the standoff distance between the facesheets was replaced by a function of the honeycomb cell size, foil thickness, and yield strength of the HC material. The BLE demonstrated excellent accuracy in predicting the ballistic limits of HCSP when tested against a new set of simulation data, with the discrepancy ranging from 1.13% to 5.58% only. The ANN was developed using MATLAB’s version 2018b Deep Learning Toolbox framework and was trained utilizing the same HCSP HVI database as that employed for the BLE fitting. A comprehensive parametric study was conducted to optimize the ANN architecture, including such parameters as the activation function, the number of hidden layers, and the number of nodes per layer. The resulting ANN demonstrated a very good predictive accuracy when tested against a set of simulation data not previously used in the training of the network, with the discrepancy ranging from 0.67% to 7.27%. Both of the developed predictive models—the BLE and the ANN—can be recommended for use in the design of spacecraft orbital debris shielding involving honeycomb-core sandwich panels.