There are Senate (and House) ethical rules regarding conflicts of interest. So the general principle being asserted is legitimate. But those rules as presently written pertain mainly to advancing or passing legislation that benefits the legislator financially. (For the same reason, Presidents or candidates have put their assets into "blind" trusts to guard against the perception they may take action knowing it will benefit them personally.)
Though, as the article notes, Senate rules at present don't clearly address the situation Cruz would face in his role on the Judiciary Committee. So there would be no actual requirement for Cruz to limit his participation.
If there were an actual case involving Cruz within the federal courts that was pending or likely to come before the SCOTUS the pressure for Cruz to recuse himself, at least from his Committee role, might be stronger. But that isn't likely to be the situation.
In the event he acquired his U.S. Passport by fraudulent misrepresentations, it can be revoked and his U.S. citizenship would become invalidated, regardless of claims about naturalized versus natural born citizenship.