Analytical solutions are derived for steady fully-developed buoyancy-driven flow in a vertical annular microchannel, of which either the inner or outer wall exhibits superhydrophobic velocity slip and temperature jump, and the inner wall is maintained either at constant wall temperature or constant heat flux. For the four possible cases of hydrodynamic and thermal boundary conditions, we determine the flow rate as a function of the core size, slip length and temperature jump coefficient. Asymptotic limits are obtained for very large slip and temperature jump. The effects of slip and temperature jump on two issues, namely, the optimum core radius for maximum flow rate and the singular increase of the flow rate for very small core radius, are investigated in particular.