We study the scattering of a polarizable atom by a conducting cylindrical wire with incoming boundary conditions, that is, total absorption, near the surface of the wire. Based on the explicit expression given recently [C. Eberlein and R. Zietal, Phys. Rev. A75, 032516 (2007)] for the nonretarded atom-wire potential, we formulate a hierarchy of approximations that enables the numerical determination of this potential to any desired accuracy as economically as possible. We calculate the complex s-wave scattering length for the effectively two-dimensional atom-wire scattering problem. The scattering length a depends on the radius R of the wire and a characteristic length beta related to the polarizability of the atom via a simple scaling relation, a = R (a) over tilde(beta/R). The "scaled scattering length" (a) over tilde tends to unity in the thick-wire limit beta/R -> 0, and it grows almost proportional to 1/R in the opposite thin-wire limit
