Precision measurements of electroweak boson production in $pp$ collisions are considered benchmark tests of the Standard Model. In particular, they provide verification for predictions of the electroweak theory and quantum chromodynamics (QCD). The rapidity of electroweak bosons produced in $pp$ collisions is strongly correlated with the initial quark dynamics. Therefore, these measurements can provide constraints on parton distribution functions (PDFs) of the proton. This study presents the first measurement of $W$ boson production in $pp$ collisions at $\sqrt{s} = 5.02$ TeV using data collected by the ATLAS experiment in 2015. Fiducial cross-sections for $W^+$ and $W^-$ boson production are measured in leptonic decay channels. The cross-sections are measured inclusively and differentially in the decay lepton pseudorapidity. In addition, a measurement of the lepton charge asymmetry is presented. The measured cross-sections and charge asymmetry are compared to predictions calculated at next-to-next-to-leading order in QCD using several recent sets of PDFs.