Would I not have to hook this up to an amplifier circuit first? Or just straight to the Arduino?
Try hooking it to the Arduino T1 input (digital pin 5) through a 10K series resistor. If that doesn't work, you can use either of the two preamplifiers shown in that page I linked to.
Another possibility is to use the analog comparator (with noise filter enabled) as the amplifier, and measure the time taken to receive a fixed number of interrupts.
As well, I wouldn't really know how to go about mapping the result to pressure, as the frequency readings I get from a dmm and the frequency counter are completely different.
You said "The sensor is based on a pressure controlled quartz crystal oscillator with frequency 36-40kHz 0-200psia.". Doesn't that give you the required mapping? Do the readings from the frequency counter agree with that?