There are several things that need to be addressed before even beginning a discussion on this topic.
It is always God's will for the gospel to be preached (Rom. 10:15, 1 Tim. 2:4, Col. 1:28, 2 Cor. 5:14-6:2). Nobody would ever be sinning by preaching the gospel. Saying that Paul was out of the will of God for going to Jerusalem is not saying that he was out of the will of God for preaching the gospel.
The "will of God" is not a mysterious thing, it is what God has revealed in His word. We can KNOW what the will of God is (Rom. 12:2, Eph. 5:17) and it is for all men to be saved (John 6:40, 1 Tim. 2:4).
Which brings me to what I believe the answer to this question is: Paul was in the will of God for preaching the gospel in Jerusalem, that was the revealed will of God (John 6:40, 1 Tim. 4:2, 2 Cor. 5:14-6:2). Paul was not in the will of God for going to Jerusalem at that time, it was contrary to His revealed will (Acts 21:4).
Here are the scriptures that say Paul was out of the will of God for going to Jerusalem....
 Save that the Holy Ghost witnesseth in every city, saying that bonds and afflictions abide me.
 But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy, and the ministry, which I have received of the Lord Jesus, to testify the gospel of the grace of God.
 And the next day we that were of Paul's company departed, and came unto Caesarea: and we entered into the house of Philip the evangelist, which was one of the seven; and abode with him.
 And the same man had four daughters, virgins, which did prophesy.
 And as we tarried there many days, there came down from Judaea a certain prophet, named Agabus.
 And when he was come unto us, he took Paul's girdle, and bound his own hands and feet, and said, Thus saith the Holy Ghost, So shall the Jews at Jerusalem bind the man that owneth this girdle, and shall deliver him into the hands of the Gentiles.
 And when we heard these things, both we, and they of that place, besought him not to go up to Jerusalem.
 Then Paul answered, What mean ye to weep and to break mine heart? for I am ready not to be bound only, but also to die at Jerusalem for the name of the Lord Jesus.
 And when he would not be persuaded, we ceased, saying, The will of the Lord be done.
(The Holy Ghost sent the prophet Agabus from Judea to warn Paul of what would happen if he went to Jerusalem. Then Luke, Philip, and the other brethren tried to talk Paul out of going to Jerusalem, but he wouldn't be persuaded.)
Why was Paul so stubborn? Why was he so zealous about preaching to the Jews? It was because Paul cared very much for his kinsman (the Jews) and wanted them to hear the gospel.
 I say the truth in Christ, I lie not, my conscience also bearing me witness in the Holy Ghost,
 That I have great heaviness and continual sorrow in my heart.
 For I could wish that myself were accursed from Christ for my brethren, my kinsmen according to the flesh:
 Who are Israelites; to whom pertaineth the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises;
 Whose are the fathers, and of whom as concerning the flesh Christ came, who is over all, God blessed for ever. Amen.
Paul had great heaviness and continual sorrow in his heart for the Jews, even to the point where he could wish that he was accursed from Christ for them. Paul wanted to get this burden off his back and go preach to them.
So to answer my friend's objections:
1.) Paul was not motivated by self-will (Acts 20:22-24; 21:13)
(Paul was motivated by his sorrow for his kinsmen, the Jews. Notice that he does not care about his own life and the he had continual sorrow in his heart for the Jews (Rom. 9:1-5). That is what motivated Paul, not the revealed will of God......"I go bound in the spirit....But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy....What mean ye to weep and to break mine heart? for I am ready...". While Paul was not being selfish in the sense of wanting to protect himself from harm, he was being selfish in only thinking of what would make him happy instead of what God's will was at that time. God was directing where he wanted Paul to preach and Jerusalem was not where He wanted Paul at that time.)
2.) He was arrested due to his OBEDIENCE (26:15-21)
(When Paul said "I was not disobedient unto the heavenly vision" the context is his initial sending that he received at his conversion in Acts 9, not the warnings of the Holy Ghost in Acts 20-21.)
(So Paul was at the Jesus Christ County Jail? I am pretty sure he was indeed at Rome and arrested. The Holy Ghost said in Acts 21:11 that Paul would be delivered to the GENTILES if he went to Jerusalem, not that he would be delivered to Jesus Christ. Paul was a prisoner of Jesus Christ when he was in Rome because he was put in prison for preaching Christ (Acts 22:1-21). What Paul was preaching was never the issue (Acts 23:11), the issue was WHERE God told him to go (Acts 21:4).)
This issue is not very important, but I am afraid that my friend idolizes the apostle Paul by insisting that Paul did not sin in the book of Acts. Not even Paul made such a claim...
 Brethren, I count not myself to have apprehended: but this one thing I do, forgetting those things which are behind, and reaching forth unto those things which are before,
 I press toward the mark for the prize of the high calling of God in Christ Jesus.
Paul was no doubt the most godly Christian that ever lived, but like all of us, he had TWO natures (Gal. 5:16-6:3, Col. 3:9-11). What Paul thought he should do in his flesh (Rom. 9:1-5, Acts 20:22-24, 21:8-14) and what the Spirit said he should do (Acts 21:4).