To spoil a child is to indulge the child's desires and whims excessively, to the extent that the child's character deteriorates.
In "to spoil someone rotten", "rotten" acts as an intensifier. Thus the phrase means "to indulge someone's whims and desires to such an extent that the person's character almost visibly deteriorates" (like an apple or a banana left in a sunny patch).
"To spoil someone rotten" is always used in a humorous sense: it's a humorous exaggeration. In your example, it simply means that the speaker intends to do everything possible to ensure that Anna has a very good time indeed on 26 March.