Always use "on." This is true not just for birthdays, but for any specific day or date: "on Thursday," "on Halloween," "on September first," "on the first day of spring," "on your next vacation day."

"We had a party on his birthday."

"She got a new bicycle on her birthday."

"We're planning to take a trip on my birthday."

Hi,

It depends on what you’re trying to say. For example, “I’m at his Birthday” that would mean that you are physically “at” a place. Like his birthday party.

An example of “On his birthday” would be something like “On his birthday, I want to give him a present.”