We use "as [someone] would be the first to admit" after saying something about that person, when we wish to express that the person themselves would agree with what we said (admit that it is true). Usually this is done when we are saying, kindly, something slightly negative about the person - e.g. my sister is sometimes lazy, as she would be the first to admit. To say this about someone's age is be strange, especially if the age is as low as 58, unless it is a joke (JD Salinger was a humorist, and this story contains some humour). I am pretty sure the joke is that the narrator's mother-in-law, in fact, keeps reminding her daughter and him that she is "not getting any younger" in order to get an invitation to visit.