Authenticate a user for current request

Yes, you can hook determine_current_user. This is how WordPress calls the existing code that processes authentication cookies:

add_filter( 'determine_current_user', 'wp_validate_auth_cookie'          );
add_filter( 'determine_current_user', 'wp_validate_logged_in_cookie', 20 );

e.g. see the implementations of those in wp-includes/pluggable.php.
Your filter should return the user ID of the user you want to authenticate as once you’ve processed the headers.

That said I assume your application server has done the authentication, so you don’t need to actually validate the password here: the username should be enough, provided there’s no way for an external user to forge the authenticate-to-WordPress header (and get that through your reverse proxy).