wordpress data in codeigniter project

All you need to do is query the WordPress tables for the blog data. I'm not going to do all of the work for you, because that wouldn't be fair if this is an assignment, but here is a start:

I initially thought the same but he didn't mention if he has access to the WP database.  That might be why he was doing the REST API, but since he mentioned the API I went that direction.

If both sites are on the same server and the assignment allows it you're entirely correct, and it's the easier way to go about it.

If that's the case the best thing to do, considering that it appears there is access to both servers, would be to set up the WP server to allow remote MySQL connections from the CI server.

True, but whitelisting remote servers for MySQL access can be a slippery slope for (assumed) new folks and open the server to security  vulnerabilities if done incorrectly... but I digress...  I don't want to get this too off topic.

Hopefully he finds one of the two ways helpful.