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.