What should I do? Should I write her a very sentimental (& honest) email to apologise and as for her forgiveness?
hate to say, but this will not bring her back.
I think you should ask yourself for what are you apologising: you told her that you were 36, while you are 39. That is absolutely nothing. If she was really into you from the beginning, this small fact should not matter. As you mentioned in your post, she said she wants to be friends and there is no sex. This is strong evidence that she likes you (otherwise she wouldnt spend time with you), but she doesnt see you as a potential (there is no sex)
She gets mad, and you accepted it. To her this means, that you are not a challenge for her anymore.
To be honest, she wasn't sure about her feelings for you. Accepting her tantrums and apologising will not help you. She was looking for a reason to stop with you.
I do sense some oneitis in your message. If she really likes her, she will come back and admits that the 3 yrs extra is not that bad.