How can I alter methods in ContentTranslationHandler? I want to alter some of the methods here:
https://git.drupalcode.org/project/drupal/blob/8.8.x/core/modules/content_translation/src/ContentTranslationHandler.php

How can I alter just some of the methods?

Or can I override this class in any way?
There are mainly two ways 

1- create a patch https://www.drupal.org/node/707484 then maybe you can apply it using composer patches.

2- override the method in your module by extending the class ContentTranslationHandler