We show that the AdS2 metric in the presence of a Maxwell field together with the constant dilaton field satisfy the equations of motion for the heterotic string theory. Then, by applying the non-Abelian T-duality (here as Poisson-Lie T-duality on a semi-Abelian double), a non-Abelian dual pair for the AdS2 metric is constructed. By using a certain parametrization of the two-dimensional non-Abelian Lie group A2 we construct the original sigma model including the AdS2 metric in the absence of a B-field. It is shown that the dual background constructed by means of the Poisson-Lie T-duality is supported by a B-field and a metric whose contains a true singularity. By examining the behavior of the dual metric in the region r=0, it is shown that the metric along with a non-trivial dilaton field and an appropriate Maxwell field satisfy the equations of motion of the heterotic string theory for very small r. The most interesting indication of this work is that for large r the dual metric approaches the asymptotic solution; in fact, the dual metric is an asymptotically AdS2 one. Finally, we discuss D-branes and the worldsheet boundary conditions defined by a gluing matrix on the AdS2 sigma model. Using the duality map obtained from the canonical transformation description of the Poisson-Lie T-duality for the gluing matrix which locally defines the properties of the D-brane, we find three different cases of the gluing matrices for the AdS2 sigma model and its dual pair.