In the present work, an axisymmetric lattice Boltzmann solver is developed to simulate blood flow through stenosed as well aneurysmal arteries. The developed solver utilizes characteristics-based finite-difference implementation in general curvilinear coordinates. Both the axisymmetric nature of the flow as well as the blood rheology are incorporated through respective source terms. Using Chapman-Enskog expansion it is verified that the proposed source terms recover the correct axisymmetric Navier-Stokes equations for both Newtonian and non-Newtonian fluids. The solver is verified for steady and pulsatile inflow for flow through an abdominal aortic aneurysm. Further, in the spatially developing pulsatile flow the various time-scales involved are optimized in order to correctly simulate the flow physics. The pulsatile blood flow through a model irregular stenosed lumen is analyzed in order to correlate the blood flow dynamics with the amplitude and frequency of irregularity. The effects of pathology (either stenosis or aneurysm) on the blood flow dynamics and disease progression are investigated using vorticity dynamics, wall shear stress, oscillating shear index and luminal surface concentration. Overall, it is noted that the solver is capable of simulating blood rheology through pathological arteries under pulsatile conditions.