It would appear that moving violations within the United States (like the speeding tickets) may result in either a civil, a misdemeanor or a criminal charge, depending on the state and the speed.
I would indeed recommend you try to follow PassKit's advise, to see if you could get an exception for being based abroad.
Otherwise, some states, like California, let you have a trial by mail (or sometimes even telephone!), where you are not required to be physically present within the courtroom to defend your case.
If the state where you got the ticket requires personal appearance to fight the ticket (e.g., if you were charged with reckless driving, including through reckless-by-speed statutes), you could hire an attorney who will represent your interests.
Note that if you decide to just pay the ticket (thus pleading guilty or no-contest), make sure that your local jurisdiction (where you have the driver licence from) would not revoke your licence upon being informed of your violation and conviction. For example, if you were to have a North Carolina licence, and you plead guilty to doing 81mph in an 80mph zone (maybe through a plea bargain, since no state is supposed to be giving tickets less than 5mph over the limit), and the court does forward the conviction to NCDOT DMV, then NCDOT DMV would have no choice but to automatically revoke your licence due to their ridiculous reckless-by-speed laws (many states have speed limits higher than what the reckless-by-speed speeds in Virginia and North Carolina are set at!).
A good resource in regards to the speed limits would be http://www.mit.edu/~jfc/laws.html.
And, as you may note, doing 85mph or above in Arizona, is automatically what is essentially a reckless driving charge (sometimes known as "reckless-by-speed" in other states that explicitly classify it as reckless driving, such as Virginia and North Carolina), which is indeed why your citation suggests that there's a criminal case against you! (Which likely would only have been civil had you been clocked at 84mph or below.)
A. A person shall not:
3. Exceed eighty-five miles per hour in other locations.
B. A person who violates subsection A of this section is guilty of a class 3 misdemeanor.
C. A person charged with a violation of this section may not be issued a civil complaint for a violation of section 28-701 if the civil complaint alleges a violation arising out of the same circumstances.
A. A sentence of imprisonment for a misdemeanor shall be for a definite term to be served other than a place within custody of the state department of corrections. The court shall fix the term of imprisonment within the following maximum limitations:
3. For a class 3 misdemeanor, thirty days.
I'm afraid you might have no choice but to hire an attorney! And for the future, pay visits to states that don't have reckless-by-speed laws on their books!