If your daughter had to be booked and fingerprinted for her first offense, then you can expect that the state will know about her first offense when she appears in court on Monday. Nevertheless, because she was not convicted of that first offense, even if they are aware of it, this would be treated as her first offense.
I don't know the merits of the case against her. It may be that with a lawyer, she can get the MIC dismissed. Even though this is not a very serious crime, criminal cases can have lifetime repercussions if handled incorrectly in court. If the case is not dismissed a typical plea offer for something like this is a diversion program. That is a special form of probation whereby a defendant is supervised, must refrain from alcohol, must pay fines, do community service, get anti-alcohol classes/treatment/counseling, depending on what's needed, and must stay out of further difficulty with the law. If she can do all that successfully, the charge against her will be dismissed and she won't have to carry a conviction on her record.
The interesting thing is that she may also be eligible for this in Tempe as well, because the Tempe matter was her first rather than her second arrest. So she's not in bad shape to come out of this without a criminal record. But despite her eligibility for the program I mentioned above, the participants in the program are selected by the prosecutor and are at his discretion. So again, it's good to have a lawyer for her because a favorable outcome may have to be negotiated, and it's best to leave plea bargaining to a pro.