We don't know
The truthful answer is that we don't know if he is a natural born Citizen in this case. The relevant passage in the U.S. Constitution from Article II, Section 1:
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
Ted Cruz was not alive at the time of the adoption of the constitution, so the relevant portion is "No Person except a natural born Citizen...shall be eligible to the Office of President". There's no definition of "natural born Citizen" in the original constitution, so we don't know what it might mean. The fourteenth amendment makes clear that anyone born within the U.S.A. "and subject to the jurisdiction thereof" is a citizen, but it doesn't say anything about those born outside.
There are arguments both ways. It would be perfectly reasonable to say that anyone who has been a citizen since birth is a natural born citizen. If so, Ted Cruz is eligible to run for President.
There is also an argument that people born outside the U.S.A. when born are not natural born citizens. This is based on a reading of what the phrase meant in the common law at the time, and on judicial precedents regarding the citizenship of people born in the Panama Canal Zone.
Note that it is currently clear that Ted Cruz and John McCain are citizens, and Cruz has been a citizen since birth (McCain was retroactively given citizenship by congressional action). What's not clear is if they are "natural born Citizen[s]" under the constitution.
Source: McManamon, Mary (2015), "The Natural Born Citizens Clause as Originally Understood", 64 Catholic University Law Review 317
Source: Chin, Gabriel J. (2008), "Why Senator John McCain Cannot Be President: Eleven Months and a Hundred Yards Short of Citizenship", 107 Mich. L. Rev. First Impressions 1
Note that it would be possible for the Supreme Court to overturn the previous court rulings and declare McCain a citizen since birth and thus a "natural born Citizen". Those decisions have mostly been moot since laws were passed giving people born to U.S.A. citizens citizenship regardless of birthplace.
Part of the problem is that there is no authorization for Congress to set the definition of who is and is not a "natural born" citizen for the purposes of who is eligible to be President. But Cruz and McCain base their arguments on laws passed by Congress. The judicial precedents actually work against them, as there is definite precedent for denying people born outside the U.S. citizenship from before the passage of those laws. The laws were passed in response to the judicial actions.
So the only thing we can say for sure is that we don't really know. A Supreme Court decision could go either way.