The Praefect is first placed on the Basilica.
From there he advanced by the institution's numbers (i.e. from the Praetorium to the temple and from the temple to the Tavern). He is not using the paths between the institutions.
Is that right?
No. He does follow the path, around the outside clockwise.
He never visits the
Tavern Temple, because the Tavern Temple doesn't have a white prefect space for him.