How to pass parameters to anonymous class?

Yes, by adding an initializer method that returns ‘this’, and immediately calling that method:

int myVariable = 1;

myButton.addActionListener(new ActionListener() {
    private int anonVar;
    public void actionPerformed(ActionEvent e) {
        // How would one access myVariable here?
        // It's now here:
        System.out.println("Initialized with value: " + anonVar);
    }
    private ActionListener init(int var){
        anonVar = var;
        return this;
    }
}.init(myVariable)  );

No ‘final’ declaration needed.