If you are going to engineer something like this, it will be most helpful if you use correct terminology. A pulley is a device that guides a belt. How will you hang a pen from a pulley?
I will pass this array as a string to the arduino and "print" out the picture in an array format from to to bottom or whatever way you like.
Assuming that the paper is white, the "printer" only needs to be concerned about the black parts of the picture.
You'll need to know where a black pixel is, and what that translates to on the paper. You'll need to command the stepper motors to move back and forth, or up and down, as appropriate, to move the "print head" over the whole area that the pixel represents.
You will not be able to pass the entire image as string to the Arduino. It does not have enough memory to hold all the picture data. Unless the picture is very small. Converting the picture to a string will not decrease its size.
This is not an impossible project, but it is not a trivial project, by any means.
There are many examples in the Exhibition section that show how much work it really is.
But, break it into steps. Make the Arduino spin a stepper motor that is connected to nothing. Then connect it to the pulley and pen assembly, and make the Arduino move the pen a fixed amount.
Repeat for the other axis. Make the Arduino move the pen to "home", wherever that is.
Now, you need to be able to move the pen up and down.
Once this all works, then you can work on the part that moves it to specific locations based on what Processing has determined needs to be drawn.