She got the letter the day she died. She got the letter on the day she died. UK English speakers often use "at" when talking about the weekend. I.e. "Chelsea will play their final at the weekend".

"She got the letter at the day she died" is NOT proper English because "AT" is not used like this. It's NEVER "AT" this day." It's ON this day. "She got the letter ON the day she died," works. "She got the letter the day she died," works fine. It just makes the sentence shorter.

she got the letter ON the day she died. 表示具体的某一天前,应该用ON ，表示具体的时间点用AT。

The second one is the correct one. I would say: She received the letter on the day that she died. 'on' not 'at'