Cruz can run for President, but Arnold can't.
https://www.law.cornell.edu/constitution/articleii
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.
http://blogs.wsj.com/law/2015/03/23/...-as-president/