Both sentences are missing something. Try this: "Sorry for not being able to come to your birthday." Or, "For not being able to come to your birthday, I blame myself alone."
The second version has a question mark but not the sense of being a question: "For ... birthday?" Try: "What do I owe you for ... birthday?"
Perhaps you needed to explain your letter by putting a note on the envelope, or the subject line of your email message. Then the fragment sets up your reason for the letter. It would be as though you are answering the question "Why do I need to apologize? "Because I missed your party."