You'll need a preamp (probably). Microphones typically put-out millivolts (depending on mic sensitivity & sound level). Preamps put-out line-level, which is around 1V. The Arduino ADC has a 5V (DC*) and a 1V (DC*) range, so line-level is appropriate.
And since it doesn't appear to be a "standard" microphone, you'll need a preamp (of some kind) if you ever want to plug-into a mixer or PA system.
There are two aspects to your project, a preamp and the audio-to-MIDI conversion. I'd recommend developing them separately. You can built & test the preamp using a line-in (Aux or Tape) on your stereo system, your soundcard (if you have a desktop with line-in), or on your TV if it has audio inputs. You can build & test the audio-to-MIDI conversion with a solo flute CD, or with a flute recording on your computer (using line-out or headphone-put into the Arduino).
:( The information on the barcus-berry website is not clear... It says "electret", and it also says piezo. Either way, it won't interface properly with a normal "pro" preamp (which will have a low-impedance balanced input with XLR connectors and optional 48V phantom power).
Piezo pickups can put-out fairly high voltage into a high impedance amp/preamp. I believe you can plug a piezo pickup for an acoustic guitar into a regular guitar amplifier.
The Arduino has very-high input impedance, so it won't drag-down the signal. With a piezo pickup, you might get enough signal into the Arduino... Maybe 100mV, maybe more... Or, maybe you'll need a preamp. A "direct box" (AKA DI box) for a guitar should work. They make DI boxes with line-outputs, and the mank DI boxes with XLR mic-level outputs for connection to a mic peamp or mixer.
Electret mics require a power supply. Sometimes they work from a 1.5V battery. The 5V power for "computer microphones" is supplied by the microphone input on a regular soundcard. Since eletret condensers are either designed to be plugged-into a soundcard, or they are desinged to pbe plugged into a preamp/mixer have an internal battery, I don't think you can buy a preamp that provides power to an electret condenser, but you can find schematics on the Net. (Regular 48V phantom power for a "studio condenser" might fry an electret.)
- The Arduino analog-to-digital converter accepts 0-5VDC. So for audio (which is AC), you need to bias the input (typically at 2.5V). Feeding negative voltages (such as the negative half of an audio waveform) into the Arduino can damage it. Plus, for what you are doing you need to "cleanly" read the whole AC waveform.