Apple's appeal will be heard by the 2nd U.S. Circuit Court of Appeals in New York. They are expected to argue that the presence of the court appointed monitor will unnecessarily impede Apple's ongoing business efforts. They have argued in the past that the monitor's mandate is too broadly written, and that by seeking to interview top executives and board members he is also being too intrusive.
Apple has an administrative stay until the hearing, which means that Bromwich won't be monitoring the company for the next while. The stay probably won't last long, because the appeals court has indicated that they will hold the hearing as soon as possible. If Apple is successful at that hearing, they will get an injunction that will delay enforcement of the antitrust ruling until after Apple files their appeal later this year.
At this point it's difficult to predict whether the appeals court will side with Apple, but I would bet that Apple is going to lose again. Court appointed monitors are not uncommon in antitrust cases, and there is not much for Apple to complain about that isn't included in the final judgement.
And as Chris Meadows pointed out over at Teleread, Apple had several chances before the ruling was handed down to make the arguments they are putting forward now, but they did not. I should think that will make it difficult for Apple to plead their case before the appeals court, but of course we will have to wait and see what the judges think.