As the first paper of a series devoted to study the old stellar population in clusters and fields in the Small Magellanic Cloud (SMC), we present deep observations of NGC 121 in the F555W and F814W filters, obtained with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope. The resulting color-magnitude diagram reaches ~3.5 mag below the main-sequence turn-off, deeper than any previous data. We derive the age of NGC 121 using both absolute and relative age-dating methods. Fitting isochrones in the ACS photometric system to the observed ridge line of NGC 121 gives ages of 11.8 ± 0.5 Gyr (Teramo), 11.2 ± 0.5 Gyr (Padova), and 10.5 ± 0.5 Gyr (Dartmouth). The cluster ridgeline is best approximated by the α-enhanced Dartmouth isochrones. Placing our relative ages on an absolute age scale, we find ages of 10.9 ± 0.5 Gyr (from the magnitude difference between the main-sequence turn-off and the horizontal branch) and 11.5 ± 0.5 Gyr (from the absolute magnitude of the horizontal branch), respectively. These five different age determinations are all lower by 2-3 Gyr than the ages of the oldest Galactic globular clusters (GCs) of comparable metallicity. Therefore we confirm the earlier finding that the oldest globular cluster in the SMC, NGC 121, is a few Gyr younger than its oldest counterparts in the Milky Way and in other nearby dwarf galaxies such as the Large Magellanic Cloud, Fornax, and Sagittarius. If it were accreted into the Galactic halo, NGC 121 would resemble the "young halo globulars," although it is not as young as the youngest globular clusters associated with the Sagittarius dwarf. The young age of NGC 121 could result from delayed cluster formation in the SMC or result from the random survival of only one example of an initially small number of star clusters.