Fiber Fabry-Perot pressure sensors exhibit high sensitivity to dynamic pressure and can accurately capture instantaneous pressure variations; thus, they are widely used in dynamic pressure measurements. Based on this capability, the present study employs an optical fiber Fabry-Perot pressure sensor to acquire and measure dynamic impact pressure. To facilitate field testing of the impact pressure and frequency response of the fiber Fabry-Perot pressure sensor, the proposed method establishes an instantaneous frequency-swept interference model and analyzes the time-varying characteristics of the interference spectrum induced by dynamic variations in the cavity length. By performing a Fourier transform on the raw interference spectrum, applying a Hamming window to retain the positive-frequency components, and subsequently executing an inverse Fourier transform to obtain the analytic signal and its phase, the dynamic cavity length can be extracted. To further obtain the sensor′s frequency-response characteristics, a Taylor expansion of the interference spectrum is conducted, and frequency components obtained from the Fourier transform are processed through addition and subtraction operations to derive the sensor′s frequency response. The entire demodulation procedure requires no Doppler error compensation; instead, the dynamic cavity length and frequency response can be rapidly retrieved solely through Fourier-transform operations, enabling a maximum demodulation rate of up to 10 MHz. Finally, the proposed demodulation model is compared with a single-wavelength intensity demodulation method. The relative frequency-demodulation errors of the two pressure sensors are less than 1% and 0.7%, respectively, while the relative pressure-demodulation errors are below 0.1% and 0.08%. Both frequency and pressure demodulation exhibit very small errors. Experimental results demonstrate that the proposed demodulation model is highly suitable for rapid measurement of impact pressure and frequency response by using fiber Fabry-Perot pressure sensors. Compared with conventional demodulation approaches, the proposed model achieves higher speed and a simpler operational procedure, making it more appropriate for field environments with harsh conditions.