The structure of the inner region of an advection-dominated accretion disk around a nonrotating black hole is explored by applying asymptotic analysis in the region just outside the event horizon. We assume that the viscous transport is described by the standard Shakura-Sunyaev prescription throughout the disk, including the inner region close to the horizon. One of our goals is to explore the self-consistency of this assumption by analyzing the causality of the viscous transport near the black hole. The effects of general relativity are incorporated in an approximate manner by utilizing a pseudo-Newtonian gravitational potential. Analysis of the conservation equations yields unique asymptotic forms for the behaviors of the radial inflow velocity, the density, the sound speed, and the angular velocity. The specific behaviors are determined by three quantities; namely, the accreted specific energy, the accreted specific angular momentum, and the accreted specific entropy. The additional requirement of passage through a sonic point further constrains the problem, leaving only two free parameters. Our detailed results confirm that the Shakura-Sunyaev viscosity yields a well-behaved flow structure in the inner region that satisfies the causality constraint. We also show that the velocity distribution predicted by our pseudo-Newtonian model agrees with general relativity in the vicinity of the horizon. The asymptotic expressions we derive provide convenient boundary conditions for the development of global models via numerical integration of the conservation equations.