Concerning the verb of 'assert'

6. 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.

and

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.

and

He applied his authority on his subordinates/countrymen/party members etc.

Apply and Implement are words unto themselves with their own definitions.