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"line management" = administration of the activities contributing directly to an organization's output. So he is in charge of the way in which the work is done.
The meaning of "line" varies from job to job.
A line manager is someone who is responsible for overseeing the performance of another, more junior, employee.
On a movie set, the line producer is responsible for administering the budget of the film.
In your sentence, therefore, it would seem to refer to the obvious duties of his position as Finance Officer, in administering the budget of the organization ('line'), and also responsibility for overseeing staff.
line 
Re: line 
Linear Mode
