The chronological window
Joseph was born in Padan-aram. Benjamin was born later, near Bethlehem, before the events of Genesis 37 in which Joseph is seventeen. Joseph must therefore have been between infancy and sixteen when Benjamin was born.
The text does not say how many years separated the two births. Any single number is an estimate, not a quoted age.
Why Joseph still knew he had a brother
Even if Benjamin was born only shortly before Joseph was sold, the narrative order allows Joseph to know of a younger brother from Rachel. When Joseph later sees Benjamin in Egypt, he recognizes him as his brother, his mother’s son, and is deeply moved.
For the reverse question—Jacob’s age at Benjamin’s birth—and for Benjamin’s age in Egypt, use the linked articles.