I have good news, and I have bad news.
The good news is that only two changes to the board are required to change it from a 5V board to a 3.3V (or any voltage within spec) board.
The bad news is that there's no way in hell you'll be able to do them.
So, we work around them and get the board up and running.
The first is obvious: a 3.3V board should be running off of 3.3V. Since we can't replace the SOT23-5 regulator on the board (the little 5-pin device towards the end) we need a voltage regulator to supply this, directly to the VCC pin. Find yourself a linear or switching regulator that provides this voltage.
The second is that the microcontroller, an Atmel ATmega168/P, is not specced to run at 16MHz when the supply voltage is only 3.3V; the maximum speed is 12MHz. Since we can't replace the crystal on the board (the little silver package next to the MCU itself) we need to perform some trickery to tell the MCU divide the frequency down to something way below safe, and then boost it up after. Use a programmer (see some of my other answers for more info about this) to set the CKDIV8 fuse and then use the routines in
avr/power.h in your code to "reduce" the divisor to 2, bringing the clock speed back up to 8MHz. While you're at it, check the BOD fuses to make sure they're not set to something useless like 0b100.
As always, see the datasheet.