Paul claimed the gospel as his or the apostles'. In another place, he puts it more directly, "that if my gospel. . . ." He is saying that he preaches exactly the same message as Jesus Christ did. In other words, there was a clear transference, into Scripture, from the message of Christ to the original twelve and then to Paul (by Christ directly; see the inference in Galatians 1:16). Paul then claims it as his own because it was now in him. He was living it and delivering it—now it was his gospel. When he preached, people were truly converted to the true faith.
In other places, people like John say basically the same thing, for instance, in I John 1:1-5. He opens the epistle by saying, "This is the message which we have heard from Him and declare to you" (verse 5). He was speaking either in the royal "we" or including the other apostles as having and preaching the same message. At the time, Gnosticism was devastating the true church because it appealed to the carnality within people, and John was attempting to steer them back to the true gospel.
John W. Ritenbaugh
The Wisdom of Men and Faith
Other Forerunner Commentary entries containing 2 Corinthians 4:3:
1 John 2:15-17