A solution method of the Holstein-Biberman equation in the case of two-dimensional finite-size geometry by means of transformation of the integral operator to a four-dimensional matrix is presented. Using this matrix the array of two-dimensional eigenvalues and eigenfunctions of the radiation transport operator in the case of finite cylinder is determined. The exact two-dimensional characteristics have been compared with approximate functions determined as a combination of corresponding eigenvalues and eigenfunctions for the one-dimensional problems (cylinder of infinite length and slab). The spatiotemporal evolution of excited atom densities for two typical forms of the excitation source in a nonequilibrium plasma has been analyzed. The reasons for the distinct difference in the formation of spatiotemporal distributions of resonance and metastable atoms in the case when the spatial distribution of the excitation source does not coincide with the fundamental mode are discussed. Resonance atoms follow the excitation source while the diffusion effectively takes metastable atoms out from the excitation source. Rearrangement of metastable atoms to the fundamental mode during their decay lasts about one effective diffusion lifetime while the corresponding process for the resonance atoms takes much longer (several effective lifetimes). The differences are caused by the effective suppression of higher diffusion modes compared with radiation modes. The developed solution method treats the radiation transport processes at the same accuracy level as diffusion transport of other plasma components and it is suitable for a self-consistent modeling of nonequilibrium plasmas.