The ironic about Mrs. Walker's name is that she rides in a carriage.
The correct answer is the first one.
Irony, let's remember, is a figure of speech in which words are used in such way that their intended meaning is different from the actual meaning of the words. It may also be a situation that ends up in quite a different way than what is generally anticipated.
In this case, it is funny / ironic that a person called Walker actually rides in a carriage instead of going places by foot.