API経由で特定の投稿を「いいね!」するにはどうすればよいですか?

Is there anything in the Discourse API that can help you like a post?

Yes, see How to reverse engineer the Discourse API and Discourse API Docs

「いいね!」 3

what is the “post_action_type_id”?

For like it is 2 as you can check via How to reverse engineer the Discourse API

「いいね!」 2