Is it possible to solve this in the software, by letting the rfid-readers listen in turns?
Yes, as long as you don't mind missing what the other device sent to the serial port while not listening to it.
and how would this be done?
Using any method of an instance of NewSoftSerial (or SoftwareSerial::listen() when you move up to 1.0).
I am currently using the Arduino Uno, which uses the 0 and the 1 port for serial communication, is it possible to use these ports?
The hardware serial port (pins 0 and 1) is how the Arduino IDE loads a new program, so the reader would need to be unplugged/powered off while uploading a sketch. Other than that, the serial port is available for debugging OR connecting an external device (such as a RFID reader) to.
I also need to send data to processing which occupies the usb port, I once read the usb port can't be used at the same time as the 0 and 1 port is this correct?
It would be better if you had mentioned ALL your requirements up front.
Seriously, it's time to get out your wallet and order a Mega.