Raman spectroscopy is among the top analytical techniques for ultra-low-dense organic matter, crucial to the search for life and analysis of celestial body surfaces in space exploration missions. Achieving the ultimate sensitivity of in-situ Raman spectroscopy necessitates a breakthrough in detecting inelastically scattered light. Single-photon detectors (SPDs) operating in photon counting mode, which can differentiate between Raman and luminescence responses, are promising candidates for the challenging scientific requirements. Since large SPD arrays are not yet commercially available, a dispersive element can be adapted to a single-pixel detector. By exploiting chromatic dispersion in optical fibers and picosecond-pulsed excitation, we delay the arrivals of different spectral components onto a single-pixel SPD. This method also separates weak Raman signals from stronger luminescence through correlated time-domain measurements. We study the impact of fiber properties and the excitation wavelength of a pulsed laser on the spectral resolution of the fiber-dispersive Raman spectrometer (FDRS). Additionally, we demonstrate the FDRS's potential for studying biomarkers and discuss its feasibility for analyzing inclusions in ice matrices.