What would the code be? How would i do it?
Firstly, write a sketch that reads the joystick position and just prints it out, to prove you can. And write another sketch that just moves the servo through a fixed sequence of positions, to prove you can.
Make sure you have a clear and complete understanding of what behaviour you want to provide. For example, in your description you have described how the output moves to the left when the joystick moves left, but does not move back to center when the joystick moves back.What else happens? Is there some way to move the output back to center? Is there any way to move it further to the left? The details matter.
Finally, write your sketch that reads the joystick position and uses it to control the servo position, using the algorithm you have described above. If you want some help on this last bit, then you will need to explain to us what the behaviour is that you're trying to provide.