Can I connect my ADMP401 straight to my computer's microphone jack?
That "microphone" is actually a complete assembly that includes a amplifier.  As the spec says, it puts out about &plusmn;200 mV when you talk normally into it from arms length.

You want to connect this to the "line in" jack of your computer, not the microphone jack.  The line in jack is meant for signals like this that are already at a roughly standard &plusmn;1 V amplitude.  The microphone input assumes signals from a bare microphone, and therefore applies lots of amplification.  The result would clip and distort with the signals from your microphone assembly.