It is not that Arminians argue "against" predestination - that would be silly, since the term comes from Scripture; Arminians would have to cross out a whole bunch of verses in their Bibles if that were the case. Clearly "predestination" is a reality.
The question is, what does that mean, and how does that work?
In general, Calvinists teach that predestination means that God, in His complete sovereignty (i.e. control over everything), predetermined who would be saved and who would spend eternity in Hell, based solely on His plan which He designed for His glory. It is not about us, what we do, what we decide (as if we had some autonomous "free-will"); it is about God's plan for Self-glorification.
In general, Arminians teach that predestination means that God, in His complete wisdom and foreknowledge (i.e. awareness of every future event), prepares the future based on His awareness of what men will choose.
Where do Arminians get such strange ideas? From Scripture!
For those whom He foreknew, He also predestined ... and these whom He predestined, He also called; and these whom He called, He also justified; and these whom He justified, He also glorified. -Romans 8:29-30
Peter, an apostle of Jesus Christ, To those ... who are chosen according to the foreknowledge of God the Father -1 Peter 1:1-2
It is easy to think that the opposing view is ridiculous, but we really need to listen more carefully to what the "other side" is saying; one's "opponents" in this debate are not as ignorant as they might seem.
From a Calvinist perspective, Arminians don't believe God is sovereign, because they don't believe He is ultimately behind every choice that is made. To them the end result is that Arminians don't believe that God is God.
From an Arminian perspective, Calvinism paints men as mechanical creatures who carry out the actions God programmed them to carry out, more or less. The end result is a God who decided that the best possible plan was one where He made billions of people for the sole purpose of torturing them in Hell for eternity - all for Self-glorification. And yet somehow "God is love".
What About Judas?
The story of Judas doesn't catch Arminians off-guard in any way; nor does it pose a challenge. Judas chose to betray the Lord, and God knew He was going to do so. Based on His foreknowledge, He arranged history accordingly, foretold it to His prophets, and then pointed to it when it happened.
All of this causes the Arminian to bow in worship to the all-knowing, all-wise Creator and Lord.
Of course, the Arminian would not concede that God wanted Judas to choose evil and inherit eternal torment, but for the Arminian, this is more of a comfort than a concern. It does not signify that God is not God; only that God is not malicious.