How to interpret spectrogram correctly?
The problem is not the spectrogram parameters, these are correct since they only depend on what resolution you want in time and frequency domain. Also, the spectrogram interpretation is correct, there are multiple frequency peaks. The problem may be:


  I expected to see one high power frequency after pressure rise, instead of multiple frequencies 


Why? If you are not in free-field conditions (and you mention a cylinder) there will be multiple wave reflections, resonances, standing waves and 3D pressure patterns. A sudden pressure step will distribute its power over the spectrum, but frequency peaks will appear because you are essentially "tuning" your cylinder. 

Recall that pressure waves travel at sound speed. Considering for example the habitual 340 m/s; if your cylinder measures 1m, the wave front will travel back and forth each 0.0029s, adding multiple frequency components to the measurement.

As for standing wave patterns, we can consider the case of a closed cylinder, where we can compute the frequency of the fundamental frequency as $f_1 = \frac{a}{4L}$ with subsequent harmonics at $nf_1$ where $n$ are odd integers (if your cylinder is indeed closed, does it have a length of ~20cm?).



Regarding the problem of measuring higher frequency waves in ducts, consider that according to Eriksson, it may be assumed that waves in a circular duct propagate in a planar fashion (i.e., pressure is constant in every cross-sectional area of the duct) only below a cut-off frequency of:
$$
f = 1.84 \frac{a}{\pi D}\sqrt{1-M^2}
$$

Where a is the sound speed, D the duct diameter and M the Mach number of the mean flow. Above this frequency, the pressure distribution in a given section of the duct will not be uniform, it will have nodes. See the following animation for an idea on how pressure varies in a given planar section, noting that only the Z(0,0) is a typical planar wave. 



Beyond the planar wave frequency it is difficult to correctly measure the pressure spectrum with only one transducer. In any case, I believe that the signal processing that you are performing is correct; the only problem is that you are witnessing the acoustic effects of your cylinder.