We use numerical N-body hydrodynamical simulations with varying PopIII stellar models to investigate the possibility of detecting first star signatures with observations of high-redshift damped Ly$alpha$ absorbers (DLAs). The simulations include atomic and molecular cooling, star formation, energy feedback and metal spreading due to the evolution of stars with a range of masses and metallicities. Different initial mass functions (IMFs) and corresponding metal-dependent yields and lifetimes are adopted to model primordial stellar populations. The DLAs in the simulations are selected according to either the local gas temperature (temperature selected) or the host mass (mass selected). We find that 3% (40%) of mass (temperature) selected high-$z$ ($zge5.5$) DLAs retain signatures of pollution from PopIII stars, independently from the first star model. Such DLAs have low halo mass ($<10^{9.6},rm M_{odot}$), metallicity ($<10^{-3},rm Z_{odot}$) and star formation rate ($<10^{-1.5},rm M_{odot},yr^{-1}$). { Metal abundance ratios of DLAs imprinted in the spectra of QSO} can be useful tools to infer the properties of the polluting stellar generation and to constrain the first star mass ranges. Comparing the abundance ratios derived from our simulations to those observed in DLAs at $zge5$, we find that most of these DLAs are consistent within errors with PopII stars dominated enrichment and strongly disfavor the pollution pattern of very massive first stars (i.e. 100~$rm M_{odot}$-500~$rm M_{odot}$). However, some of them could still result from the pollution of first stars in the mass range [0.1, 100]~$rm M_{odot}$. In particular, we find that the abundance ratios from SDSS J1202+3235 are consistent with those expected from PopIII enrichment dominated by massive (but not extreme) first stars.
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