Lattice models have been used to simulate mass transport to predict durability of cementitious materials. In particular, the use of dual lattice meshes allows for the coupling of fracture and transport processes, which commonly occur at the same time in these materials. Literature has shown good agreement between simulations and experimental results. Nevertheless, work regarding relevant computational aspects of the numerical model are scarce. In this study, a Voronoi-discretized lattice model is used to simulate unsaturated moisture transport in cement-base materials through the Richards equation. First, investigations regarding the choice of elemental volume approximation, time-stepping procedure and quadrature are evaluated. After validation of the approximations, simulated moisture transport in sound concrete was compared to experiments and mesh and time step sensitivity were discussed. A new approach to model capillary absorption of water in cracked concrete was also proposed and its advantages with respect to existing approaches are discussed by comparing to experimental measurements. The results confirm that the model can accurately predict the transport processes for the earlier stage of capillary absorption. Furthermore, moisture ingress in cracked concrete is simulated for different crack configurations and the use of different approaches is suggested accordingly. Finally, guidelines regarding the approximations used for optimization of the computations are presented.