We have obtained Keck spectra for 16 globular clusters (GCs) associated with the merger remnant elliptical NGC 1052, as well as a long-slit spectrum of the galaxy. We derive ages, metallicities and abundance ratios from simple stellar population models using the recently published methods of Proctor & Sansom, applied to extragalactic GCs for the first time. A number of GCs indicate the presence of strong blue horizontal branches that are not fully accounted for in the current stellar population models. We find all of the GCs to be ~13 Gyr old according to simple stellar populations, with a large range of metallicities. From the galaxy spectrum we find NGC 1052 to have a luminosity-weighted central age of ~2 Gyr and metallicity of [Fe/H]~+0.6. No strong gradients in either age or metallicity were found to the maximum radius measured (0.3re~= 1 kpc). However, we do find a strong radial gradient in α-element abundance, which reaches a very high central value. The young central starburst age is consistent with the age inferred from the HI tidal tails and infalling gas of ~1 Gyr. Thus, although NGC 1052 shows substantial evidence for a recent merger and an associated starburst, it appears that the merger did not induce the formation of new GCs, perhaps suggesting that little recent star formation occurred. This interpretation is consistent with `frosting' models for early-type galaxy formation.