The increasing popularity of time-resolved X-ray absorption measurements for understanding dynamics in molecular and material systems has led to many advances in table-top sources for pulsed X-rays. We report on a table-top laser-produced plasma (LPP) source that can perform soft X-ray (SXR), near-edge X-ray absorption fine structure (NEXAFS) measurements using a laser source with 23 ps pulse duration. The spectrometer's key specifications, such as brilliance, resolution, and stability, are characterized against the more commonly used longer-pulse-duration LPP sources. The 23 ps laser produced approximately an order of magnitude weaker SXR flux than the 8 ns laser due to the smaller total energy absorbed by the plasma. The increased repetition rate, as well as the use of a high line-density X-grating, and a self-referencing scheme still allowed for NEXAFS measurements of Si3N4 and TiO2 thin films with 2.5 minute acquisition times, a resolving power of E/ΔE = 424, and a signal-to-noise ratio of 100. It was observed that degradation of the gas jet nozzle led to long-term instability of the source, which can be remediated using alternative nozzle designs. This work demonstrates the feasibility of achieving higher temporal resolution in future time-resolved X-ray absorption measurements using table-top LPP sources.