If you want to push an entity programmatically rather than through the UI, you can use the following snippet:
\Drupal\cms_content_sync\PushIntent::pushEntity($entity,\Drupal\cms_content_sync\PushIntent::PUSH_MANUALLY,\Drupal\cms_content_sync\SyncIntent::ACTION_CREATE);
This assumes that you have at least one Flow configured to push the given entity type + bundle manually. It will return TRUE if the entity has been pushed. Please note that returning FALSE does not necessarily mean that an error occurred e.g. if the content hasn't changed since the last successful push.