Recent developments in microscale heat exchangers have heightened the need for the understanding of fluid flow and heat transfer in a microchannel. In this study, we look into fully-developed buoyancy-driven flow in a vertical parallel-plate microchannel, which has one wall exhibiting superhydrophobic slip and temperature jump, and another wall being a normal no-slip surface. Analytical solutions are derived for free convection in the channel, where the heating is applied to either one of the two walls, and by either constant wall temperature or constant heat flux. We examine how the superhydrophobic slip and temperature jump may affect the volume flow rate and the Nusselt number under various heating conditions. There exists a critical value of the temperature jump coefficient, above which the flow rate will be larger by heating the no-slip surface than by heating the superhydrophobic surface, whether by constant wall temperature or by constant heat flux. The opposite is true when the temperature jump coefficient is below the critical value. Also, the temperature jump can have a negative effect on the flow rate when the heating is by constant temperature on the superhydrophobic side of the channel, but will have a positive effect when the heating is on the no-slip side of the channel.