In this article an analysis is presented to obtain the similarity solution of the steady boundary layer flow and heat transfer of a viscous fluid flowing through a vertical cylinder that is stretching exponentially along its surface. The governing partial differential equations along with the boundary conditions are reduced to into system of nonlinear ordinary differential equations by using the boundary layer approach and a suitable similarity transformation. The resulting coupled system of equations subject to the appropriate boundary conditions is solved with the help of powerful numerical technique, the Kellerbox method. The effects of the involved parameters such as Reynolds numbers, Prandtl numbers and the natural convection parameter are presented through sketches. The associated physical properties on the flow and heat transfer characteristics that is the skinfriction coefficient and Nusselt numbers are presented for different parameters.