The bible tells us Matthew’s other name was Levi which leads us to believe he was a Levite. Were the Levites deported with the ten northern tribes? Did the Levites continue their priestly service after the northern deportation?
The Levites were dispersed throughout both the northern and southern kingdoms (Numbers 35, Joshua 21), and yes, their service continued and was restored when the Temple was rebuilt (cf the book of Ezra).
I don't think the fact that his given name was "Levi" suggests that he was of the tribe of Levi.