Ultraviolet spectra of α Tau (K5 III) obtained by the Hubble Space Telescope (HST) show many emission lines affected by broad absorption from the strong wind of this red giant star. For the Mg II h and k lines there is also a narrow absorption feature in the midst of the wind absorption that has been interpreted as being from α Tau's wind-interstellar medium (ISM) interaction region (i.e., its ``astrosphere''). We try to reproduce this absorption using hydrodynamic models of the α Tau astrosphere, which show that stellar wind material heated, compressed, and decelerated at the wind's termination shock (TS) can produce significant absorption at about the right velocity. By experimenting with different model input parameters, we find that the parameter that the absorption is most sensitive to is the ISM pressure, which determines the location of and therefore the density at the TS. However, the models underestimate both the amount of deceleration at the TS and the amount of absorption for realistic input parameters. We demonstrate that these problems can in principle be resolved by modeling the TS as a radiative shock. However, a cooling timescale short enough to affect the postshock flow is only attainable if α Tau's wind speed is increased from the 27-30 km s-1 values derived from fits to wind absorption to at least 35 km s-1. The models also seem to require a very high ISM pressure of P/k~30,000 cm-3 K to induce densities at the TS high enough to yield sufficient radiative cooling. This pressure is at least a factor of 2 higher than other estimates of ISM thermal pressure within the Local Bubble. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.