If he accepted a plea agreement, then he can file a Petition for Post Conviction Relief. It must be filed within 90 days of the day he was sentenced. The court bailiff would have given him a sheet of paper with information about it when he was sentenced. If the Post Conviction Relief petition is granted, the remedy is to send the case back to the trial court and basically do the case over.
If he was found guilty at trial, then he has a right of direct Appeal. That has to be filed within 20 days of the day he was sentenced.
The sentencing laws in Arizona are extremely harsh. Possession of Methamphetamine in Arizona carries a mandatory sentencing range of 5 to 15 years in prison, flat time. It is not probation eligible.
Your comment about there being other people in the jail who are even guiltier, but receive lighter sentences, is something I sympathize with, but it is the reality of the criminal justice system. There are a large number of factors that contribute to a person's sentence. Typically a person who is caught red handed and/or confesses to the crime will receive a harsher punishment. The reason is because the government will have a very easy time convicting the person at trial, therefore the government can, and does, offer a harsher plea agreement. On the other hand, if a person has an excellent chance of prevailing at trial, the government might make a much more lenient plea offer.
I noticed that you said that your husband is "an illegal." I'm aware that it is en vogue these days with media and politicians to call people "illegals." But "illegal" is a status, a state of affairs--not a human being. I reject the dehumanization of human beings based on the crime of crossing the border illegally. The punishment they receive for their crimes are plenty severe. I try to remind myself every day that we are all children of God.
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I don't know exactly the situation he is in. Are you saying that he had a trial and lost, was found guilty. If he lost at trial then he can appeal. If he plead guilty to some crime and was sentenced within the normal guidelines for the crime that he plead guilty to then NO he can't appeal. Comparison with some other inmate (illegal or not) is never a yard stick for a sentence. Talk to his attorney for more details.