Yes you can program it with a serial port or even a USB to serial convertor but you will need to make some modifications to the design.
I would suggest you build the Serial Com Arduino like this one on your breadboard http://www.arduino.cc/en/Main/ArduinoBoardSerialSingleSided
There is no reason why you couldn't use this with a usb to serial adapter if your PC doesn't have a COM port(because all that the FT232RL IC does is act as a USB to serial convertor).
Just be careful because the signal coming from a serial port runs at "RS 232" levels which means up to 24V. The transistors in the above circuit do the level shifting, a safer option would be to possibly use somethign like the MAX232 which you will need to go a do some research on.
I haven't tried doing it this way but in theory it should work.