There's no real difference in meaning between the four, and which one you should use is generally based on set phrases.
"Way" is the most common, and you should use it when you're not sure.
"Method" is mostly used when naming a way of doing something. You might solve a maths problem with the "grid method", for example.
"Measure" and "approach" aren't used very often at all. You might say "approach" instead of "way" in formal speech, but generally it sounds unnatural.