Does Paul always state the gospel as, 'If you believe Christ died for your sins and was buried and rose again, thou shalt be saved'?
No. Paul most often states the gospel as believing that Christ died for sins, was buried, and rose again, and thou shalt be saved. But he does not always use that exact wording. In Acts, he preached the preaching of the cross and Christ crucified, God's power to those saved.
Les teaches that most of the time Paul will say that if you believe Christ died for your sins and was buried and rose again, thou shalt be saved, and so forth. But he won't always, like here in Acts. What he preached to this deputy was the same thing: the preaching of the cross. To those who perish it is foolishness, but to us who are saved it is the power of God. Paul refused to preach signs and miracles for the Jews or philosophy for the Greeks; he preached Christ crucified. That was his theme, and it pops up throughout all his letters. In Acts 13, the doctrine from Paul's lips was how Christ died for the sins of the world.