We report the response of the well-known Shockley-type surface state, which resides around $\overline{\mathrm{Y}}$ on the $\mathrm{Cu}(110)$ surface, to the presence of a lateral one-dimensional (1D) superlattice. This grating consists of alternating stripes of reconstructed $\mathrm{Cu}(110)(2\ifmmode\times\else\texttimes\fi{}1)\mathrm{O}$ and unreconstructed $\mathrm{Cu}(110)$ and can be prepared by oxygen dosing over a wide range of stripe widths and distances, respectively. Using high-resolution angle-dependent photoemission at room temperature, we study the variation of binding energy, effective mass, linewidth and energy splitting induced by the confinement perpendicular to the $(2\ifmmode\times\else\texttimes\fi{}1)\mathrm{O}$ stripes. We demonstrate that the surface state electrons on the striped $\mathrm{Cu}\text{\ensuremath{-}}\mathrm{O}$ surface show confinement properties and photoemission spectra in almost complete analogy to the $\overline{\ensuremath{\Gamma}}$-surface state electrons on stepped $\mathrm{Cu}(111)$ and $\mathrm{Au}(111)$ surface [Mugarza et al., Phys. Rev. Lett. 87, 107601 (2002)]. At low oxygen coverages our data differ from those of an earlier photoemission study of the same system performed with the sample at $100\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ [Bertel and Lehmann, Phys. Rev. Lett. 80, 1497 (1998)], which reports the observation of singularities in the density of states of quasi-1D subbands. However, the two studies agree within experimental error limits for the higher oxygen coverages. We explain the apparent difference by the temperature-induced transition from coherent emission out of 2D superlattice bands to incoherent emission from decoupled 1D quantum well states.