I can't access my Arduino Micro's USB programming port when it's in my project. I want to reroute the RX and TX pins outside to a serial-to-usb adapter.

I have this on hand from a while ago. I found the RX and TX pins in the serial connector and I connected them to the RX and TX pins of the Arduino as well as GND but when I try and program the device nothing happens. I usually get an open port error or something like that but the programming LED on the adapter doesn't light up. Conversely if I try using the Basic Stamp software I at least get a programming LED to light up on the adapter.

Do I need an FTDI breakout instead? Or am I wiring something wrong?