It appears that the narrator promises to send Da-duh a postcard of the Empire State Building because she wants to prove that her assertion is right, namely that the Empire State Building is even taller than the royal palm in the gully.
The narrator tells Da-duh that the Empire State Building is the tallest building in the world and that it has over a hundred floors. She knows this because her class visited this New York City attraction a year ago, and she went all the way to the top of the building. Not content with this last statement, the narrator presses her point home with the assertion that the Empire State Building is even taller than Bissex Hill, which they visited the other day. This is when she promises Da-duh that a postcard of the Empire State Building would show her the truth.
The narrator tells us that the drive goes out of Da-duh after hearing this. She lowers the hand that had been poised to give the narrator a slap. Da-duh does not want to believe her grand-daughter's assertions; she has always been content with the agricultural world of Barbados and does not want the old ways to be replaced with a confusing modernization that is both different and frightening to her. Thus, she is beset by 'an ancient abstract sorrow' which depresses her spirit.