Having difficulty uploading sketch to an Arduino Pro Mini
You need to hook it up with the Arduino TX --> FTDI RX, and vise versa, so crossed over.  If you take the FTDI and hook it up to the Ardiono pin to pin as you have it pictured it should work, provided the FTDI works.

No to your note about not having a serial port, you should have that as soon as you plug the FTDI into your computer, the Arduino is not necessary for this.  First things first, check you drivers, and check your USB connection.  You won't get anywhere until that Serial port is recognized on your computer.