The inherent asymmetry of the electric transport in graphene is attributed to Klein tunneling across barriers defined by pn interfaces between positively and negatively charged regions. By combining conductance and shot noise experiments, we determine the main characteristics of the tunneling barrier (height and slope) in a high-quality suspended sample with Au/Cr/Au contacts. We observe an asymmetric resistance Rodd=100-70Ω across the Dirac point of the suspended graphene at carrier density |nG|=(0.3-4)×1011cm-2, while the Fano factor displays a nonmonotonic asymmetry in the range Fodd∼0.03-0.1. Our findings agree with analytical calculations based on the Dirac equation with a trapezoidal barrier. Comparison between the model and the data yields the barrier height for tunneling, an estimate of the thickness of the pn interface d