...A former Marine was arrested trying to bring an antique rifle into Mexico.  He declared it to U.S. customs and was given paperwork to show Mexican officials.  When he took it to Mexican customs, he was arrested and thrown in one of their hellhole jails.

Of course the charge is bogus and he should be released.  Fox is beating the drum on this story and Bill O'Reilly is laying it at the feet of Secretary of State Clinton, demanding that she take immediate action to get him released.  Expect four years of negative attacks by Fox blaming Clinton for everything even remotely connected to her.

...the guy (Jon Hammar) out fast enough.  Fiscal cliff, who cares, get this guy out of a Mexican jail.

And if he's released, O'Reilly and Fox will take the credit.  Never mind that Sen. Bill Nelson made a speech about the situation on the floor of the Senate.

How can you hold an American official responsible for the actions of Mexican Authorities?

He should have checked the Mexican Laws concerning firearms.  
A google search would have told him that Mexico does not allow any firearms.  
Other than a diplomatic request, The US does not have the authority to get the release of someone in Mexico.  I don't think it would be good policy to interfere in the legal system of other nations, for every American that violates the law in other countries.

"American Exceptionalism." That is loosely defined by FOX and the right as, we can do anything we want, even if it goes against the laws and sovereignty of other countries.