In this paper we present a thorough discussion about the photometric redshift (photo-z) performance of the Southern Photometric Local Universe Survey (S-PLUS). This survey combines a 7 narrow + 5 broad passband filter system, with a typical photometric-depth of r$sim$21 AB. For this exercise, we utilize the Data Release 1 (DR1), corresponding to 336 deg$^{2}$ from the Stripe-82 region. We rely on the texttt{BPZ2} code to compute our estimates, using a new library of SED models, which includes additional templates for quiescent galaxies. When compared to a spectroscopic redshift control sample of $sim$100k galaxies, we find a precision of $sigma_{z}<$0.8%, $<$2.0% or $<$3.0% for galaxies with magnitudes r$<$17, $<$19 and $<$21, respectively. A precision of 0.6% is attained for galaxies with the highest texttt{Odds} values. These estimates have a negligible bias and a fraction of catastrophic outliers inferior to 1%. We identify a redshift window (i.e., 0.26$<z<$0.32) where our estimates double their precision, due to the simultaneous detection of two emission-lines in two distinct narrow-bands; representing a window opportunity to conduct statistical studies such as luminosity functions. We forecast a total of $sim$2M, $sim$16M and $sim$32M galaxies in the S-PLUS survey with a photo-z precision of $sigma_{z}<$1.0%, $<$2.0% and $<$2.5% after observing 8000 $deg^{2}$. We also derive redshift Probability Density Functions, proving their reliability encoding redshift uncertainties and their potential recovering the $n(z)$ of galaxies at $z<0.4$, with an unprecedented precision for a photometric survey in the southern hemisphere.
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