The High Optical Resolution Spectrograph (HORuS) is a new high-resolution echelle spectrograph available on the 10.4-m Gran Telescopio Canarias (GTC). We report on the first HORuS observations of a transit of the super-Earth planet 55 Cnc e. We investigate the presence of Na I and Hα in its transmission spectrum and explore the capabilities of HORuS for planetary transmission spectroscopy. Our methodology leads to residuals in the difference spectrum between the in-transit and out-of-transit spectra for the Na I doublet lines of (3.4 ± 0.4) × 10-4, which sets an upper limit to the detection of line absorption from the planetary atmosphere that is one order of magnitude more stringent that those reported in the literature. We demonstrate that we are able to reach the photon-noise limit in the residual spectra using HORuS to a degree that we would be able to easily detect giant planets with larger atmospheres. In addition, we modelled the structure, chemistry, and transmission spectrum of 55 Cnc e using state-of-the-art open source tools.