On a US Airways flight, I was answering the in-flight Cranium
questions. One question came up, and I think it's wrong. The question
was... True of False. If a month starts on a Sunday, then the 13th
will always be on a Friday. I said, "True." However, the question was
false. How could this be?

It can't.  

You're absolutely right, of course.  If the 1st of the month is
Sunday, the 13th must be on a Friday.  There are no two ways about it.

1 - Sun
2 - Mon
3 - Tues
4 - Wed
5 - Thu
6 - Fri
7 - Sat
8 - Sun
9 - Mon
10 - Tues
11 - Wed
12 - Thu
13 - Fri

Here are some Google search results that reiterate this obvious fact:
://www.google.com/search?hl=en&q=%22month+*+on+a+sunday%22+%22Friday+the+13th+%22+&btnG=Google+Search

--------
search strategy:
"month * on a sunday" "Friday the 13th " 

I hope this helps.

But what if the flight crossed the International Date Line?

Nice try, Probo! :-)
Of course, when you cross the International Date Line, you'll end up
ahead in Monday the 2nd or back in Saturday the 31st (or 30th, etc.,
in certain months). Either way, though, the 13th of the month in which
you began will still end up on a Friday.

You can't trust USAir since it was acquired by Southwest.

If you consider  the adjustment  to the gregorian calander:
The day after Thursday October 4th 1582 became Friday October 15th,
When Pope Gregory cut 10 days off the year to account for precession,
and cause fall and spring equinoxes to occur at the proper times of
each season.  On this month, the 13th did not fall on a friday, in
fact it did not occur at all.