We present a coarse-grained model for stochastic transport of noninteracting chemical signals inside neuronal dendrites and show how first-passage properties depend on the key structural factors affected by neurodegenerative disorders or aging: the extent of the tree, the topological bias induced by segmental decrease of dendrite diameter, and the trapping probabilities in biochemical cages and growth cones. We derive an exact expression for the distribution of first-passage times, which follows a universal exponential decay in the long-time limit. The asymptotic mean first-passage time exhibits a crossover from power-law to exponential scaling upon reducing the topological bias. We calibrate the coarse-grained model parameters and obtain the variation range of the mean first-passage time when the geometrical characteristics of the dendritic structure evolve during the course of aging or neurodegenerative disease progression (a few disorders for which clear trends for the pathological changes of dendritic structure have been reported in the literature are chosen and studied). We prove the validity of our analytical approach under realistic fluctuations of structural parameters by comparison to the results of Monte Carlo simulations. Moreover, by constructing local structural irregularities, we analyze the resulting influence on transport of chemical signals and formation of heterogeneous density patterns. Because neural functions rely on chemical signal transmission to a large extent, our results open the possibility of establishing a direct link between the disease progression and neural functions.