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My birthday next year falls on a Sunday.
Literal
Next-year [possessive-の] my [possessive-の] birthday [topic-は] Sunday [to-に] corresponds.
当たる ('to hit / correspond / fall on') has many senses; here it's the calendar sense 'fall on [a particular day/date].' The noun phrase 来年の僕の誕生日 chains two possessives: 'next-year's my birthday.' 僕 is a masculine first-person pronoun, relatively casual.