means 'finally', in the context of that something has been a long journey/taken a lot of effort, and you have finally reached the end.
とうとうお前が大学生か…so, you're finally a college student
(You have finally reached the end of high school.)
到頭夏休みだ It's finally summer break! (It's finally the end of the school year!)
とうとう夏休みが終わるひが来た。 The day that summer break ends has finally arrived (It's finally the end of the summer break.)