We present a proof of principle experiment on single-shot near edge soft X-ray fine structure (NEXAFS) spectroscopy with the use of a laboratory laser-plasma light source. The source is based on a plasma created as a result of the interaction of a nanosecond laser pulse with a double stream gas puff target. The laser-plasma source was optimized for efficient soft X-ray (SXR) emission from the krypton/helium target in the wavelength range from 2 nm to 5 nm. This emission was used to acquire simultaneously emission and absorption spectra of soft X-ray light from the source and from the investigated sample using a grazing incidence grating spectrometer. NEXAFS measurements in a transmission mode revealed the spectral features near the carbon K-α absorption edge of thin polyethylene terephthalate (PET) film and L-ascorbic acid in a single-shot. From these features, the composition of the PET sample was successfully obtained. The NEXAFS spectrum of the L-ascorbic acid obtained in a single-shot exposure was also compared to the spectrum obtained a multi-shot exposure and to numerical simulations showing good agreement. In the paper, the detailed information about the source, the spectroscopy system, the absorption spectra measurements and the results of the studies are presented and discussed.