Is there any way that I can pass a function as a json string (conversion with JSON.stringify), send it to another function, parse the json and then execute the function that
Yes, you can convert a function to a string with it's toString() method.
Here's an example to show converting a function to a string and back to a function:
var myfunc = function () {
alert('It works!');
}
var as_string = myfunc.toString();
as_string = as_string.replace('It works', 'It really works');
var as_func = eval('(' + as_string + ')');
as_func();