The zeroth, second and fourth frequency sum rules of the transverse stress auto-correlation function of Rb have been evaluated for six thermodynamic states along the liquid - vapour co-existence curve by using the Ashcroft pseudo-potential and the corresponding pair correlation function obtained by molecular dynamics simulation. The numerical results for the sum rules thus obtained and a model for the memory function appearing in the Mori formalism are used to calculate the shear viscosity. The results obtained have been used to compare with the experimental results and with those of Kahl and Kambayashi. It is found that our method provides almost quantitative explanation for the density and temperature dependence of the shear viscosity of expanded rubidium.