Composite lattice shells are widely employed across various industries due to their exceptional strength-to-weight ratio and ability to sustain significant loads while remaining lightweight. Buckling analysis and the accurate prediction of the critical buckling load are among the most essential design considerations for these structures. In this study, the influence of incorporating carbon nanoparticles (1, 1.5, 3, 4, and 5 wt%) on the elastic properties of the resin was investigated using the Modified Halpin–Tsai model. The effective properties of the composite material in the conical lattice structure were then estimated through the rule of mixtures. Based on these properties, finite element models were developed in Abaqus to predict the buckling load, and the numerical results were validated against experimental data. Furthermore, to enable neural network–based prediction of the critical buckling load, 75 parametric models were generated in Abaqus with varying nanoparticle concentrations and aspect ratios (0.5, 1, and 1.5) for the cross-sections of helical and circumferential ribs. The corresponding critical buckling loads were calculated and used to train neural networks. Networks with different neuron counts and training epochs were systematically evaluated, and the optimal architecture for accurate prediction of the critical buckling load was identified.