- to assert oneself OED
to insist upon the recognition of one's rights or claims, and take
means to secure them.
From this sense of assert, your sample sentences could read
He asserted himself on his subordinates/countrymen/party members etc.
A good librarian is able to assert her authority when required.
going as far as necessary to accomplish such.
Naval forces can assert their authority by war. Police can assert their authority by arresting you. As far as being synonymous with apply or implement, no, but those words could be used, as in:
A good librarian is able to implement his/her authority when required.
He applied his authority on his subordinates/countrymen/party members
Apply and Implement are words unto themselves with their own definitions.