How do Arminians understand the doctrine of Unlimited Atonement in a way that does not make God unjust? The doctrine of unlimited atonement teaches that Christ died for all mankind, but only paid for the sins of those who believe in Him.  If this is the case then either Jesus already knew who would believe and paid only for their sins (which sounds like limited atonement), or he paid for everyone's sins, but those people who do not believe are still viewed as guilty before God and are still punished. Hence, their sin receives a double payment. This though would seem to make God unjust (which I know He's not.)

Let me give an example why I say this.  Let's say there was a man who got a speeding ticket and stood before the judge.  The judge tells the man that the fine is $200.00 or one week in jail.  The man says he doesn't have the money, but before the man is escorted to his cell, another man comes forward and pays the $200 instead.  The fine is paid.  Now what if the judge, having accepted the $200, still sent the man to jail.  Wouldn't that make the judge unjust and wouldn't it make the person who paid the $200 feel used?  In short it would mean the judge was demanding double payment for the penalty.

Because unlimited atonement is an Arminian teaching, I would like answers from Arminians.
The Wesleyan Arminians recognised the problem implied by your analogy and resolved it by proffering the Governmental Theory of Atonement.   

Applying this resolution to your analogy

Christ has 'made' the payment at the cross - and He can't unmake the payment because the cost to Him was real) - but rather than a completed individual transaction on behalf of each sinner (the penal substitution theory of atonement), it is like setting up a massive trust fund which has enough for all but must be applied effectually for each case.   

Analogy as an Answer

The individual application is like writing a check against the trust fund for this amount and presenting this to the individual concerned (i.e. preaching the gospel to them).  They can receive or reject this offer.  Furthermore, after receiving, they can still tear up the check or lose it through negligence (again this is according to Wesleyan Arminianism).  They will receive the check, hang on to it and present it when it falls due if they identify as Christ's (i.e. put their trust in Him, confess Him as Lord and continue to walk in obedience to Him).  Actual payment falls due at the judgment where those found outside of Christ have either rejected their checks, lost them or effectively torn them up through their unbelief.