Could somebody show me a quick example how to sort an ArrayList alphabetically in Java 8 using the new lambda syntax.
Lambdas shouldn't be the goal. In your case, you can sort it the same way as in Java 1.2:
Collections.sort(list); // case sensitive
Collections.sort(list, String.CASE_INSENSITIVE_ORDER); // case insensitive
If you want to do it in Java 8 way:
list.sort(Comparator.naturalOrder()); // case sensitive
list.sort(String.CASE_INSENSITIVE_ORDER); // case insensitive
You can also use list.sort(null) but I don't recommend this because it's not type-safe.