The present work evaluates the transport properties of thermoplastic R-BAPB polyimide based on 1,3-bis(3,3′,4,4′-dicarboxyphenoxy)benzene (dianhydride R) and 4,4′-bis(4-aminophenoxy)biphenyl (diamine BAPB). Both experimental studies and molecular dynamics simulations were applied to estimate the diffusion coefficients and solubilities of various gases, such as helium (He), oxygen (O2), nitrogen (N2), and methane (CH4). The validity of the results obtained was confirmed by studying the correlation of the experimental solubilities and diffusion coefficients of He, O2, and N2 in R-BAPB, with their critical temperatures and the effective sizes of the gas molecules, respectively. The solubilities obtained in the molecular dynamics simulations are in good quantitative agreement with the experimental data. A good qualitative relationship between the simulation results and the experimental data is also observed when comparing the diffusion coefficients of the gases. Analysis of the Robeson plots shows that R-BAPB has high selectivity for He, N2, and CO2 separation from CH4, which makes it a promising polymer for developing gas-separation membranes. From this point of view, the simulation models developed and validated in the present work may be put to effective use for further investigations into the transport properties of R-BAPB polyimide and nanocomposites based on it.