Learn French: Il a fait face à son patron.

The phrase faire face à is a fixed expression meaning to face, to confront, or to stand up to. Just like the verbs you are learning that require à before an infinitive, this verbal expression requires à to connect to the noun it targets. You cannot leave it out and say faire face son patron.

Does patron mean the same thing as the English word patron?

No, they are false friends. In French, patron usually means a boss or manager. If you want to talk about a patron of a restaurant or store, you would use a word like client.

If I wanted to say 'He stood up to him,' how would I replace à son patron with a pronoun?

You would use the indirect object pronoun lui, placing it before the conjugated verb to make Il lui a fait face. Because the expression uses à followed by a person, the indirect object pronoun is the correct choice here.

Why doesn't the past participle fait change its spelling to agree with anything here?

This verb is in the passé composé and uses the auxiliary avoir (a). With avoir, the past participle (fait) never agrees with the subject. It only agrees if there is a direct object placed before the verb, but this sentence has an indirect object introduced by à.