We present a quantitative spectral analysis of five very massive stars in the Arches cluster, located near the Galactic center, to determine stellar parameters, stellar wind properties and, most importantly, metallicity content. The analysis uses a new technique, presented here for the first time, and uses line-blanketed NLTE wind/atmosphere models fit to high-resolution near-infrared spectra of late-type nitrogen-rich Wolf-Rayet stars and OfI+ stars in the cluster. It relies on the fact that massive stars reach a maximum nitrogen abundance that is related to initial metallicity when they are in the WNL phase. We determine the present-day nitrogen abundance of the WNL stars in the Arches cluster to be 1.6% (mass fraction) and constrain the stellar metallicity in the cluster to be solar. This result is invariant to assumptions about the mass-luminosity relationship, the mass-loss rates, and rotation speeds. In addition, from this analysis, we find the age of the Arches cluster to be 2-2.5 Myr, assuming coeval formation.