I would suggest not to repeat the data on several tables. If the staff is linked to a department, define a foreign key relation from Staff to Department and have the department name only in the department table. When fetching the data, join both department and staff in order to get data rom both tables.
However, if you need to update both tables, a very traditional approach is to create a stored procedure which would contain the updates. The stored procedure would have the input values as parameters.
Have a look at
CREATE PROCEDURE[
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