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Galaxy age determination inherently deals with two questions: how much time passed during the light's travel to us, and how long before that did the galaxy form. A galaxy might be "old" in the sense that it formed billions of years ago, yet be "young" in the sense that what we are seeing is only millions of years old.
Cosmological redshift allows us to determine the light-travel time, and any spectral feature that we can recognize reliably reveals it; for example, the Balmer jump is sometimes clearly evident. However, for very distant, dim galaxies, the only spectral information available may be color indices from photometry since spectrography requires considerable telescope time and only a limited number can be so-analyzed, and if the galaxy is too dim, spectrography takes even longer, and may not be possible. In this latter case, a most-likely redshift can be determined from the observation information available. The population of galaxies seen at high redshifts show differences from the current population, which is of research interest.
Regarding the age of the galaxy producing the light we are viewing, in the nearest galaxies (including the Milky Way), observation of individual stars and stellar clusters tell us a lot: stellar age determination reveals a minimal age of the galaxy. At between-distances, the color indices and their distribution through the galaxy reveal the presence and general location of older stars and higher metallicities (metallicity is presumed to develop over time as generations of stars add more metals), thus helping pin the galaxy's likely age-range. Alpha enhancement is similarly useful. For high redshifts, an obvious limit on how long the galaxy could have existed is given by the age of the universe.