You have three options as I see it.
Cut the end off a USB cable and either solder it directly to the PCB (preferred), or solder on a USB connector with a pinout matching your board (dangerous - don't use the cable for anything else!).
Get the board remade with the correct pinout.
Solder on the USB connector upside down. Yes, upside down. If you solder the shell on to the board, you can then use magnet wire (very thin wire) to connect the data/power/ground pins to the board. The reason for soldering upside down is that your connector then has a 1:1 mapping with the board (flipping over mirrors the pins). You may also be able to bend the pins of the connector down to reach your board (no wire needed) if you can find a suitable connector.
Option 2 is the best option long term, and I would do this anyway regardless of what you do in the short term.
Option 1 is ok as long as you solder the wire on to the board - wiring up a USB connector in reverse to plug in to your board asking for trouble.