When Paul Spoke of "this ministry" in 11 Corinthians 4:1 he was speaking of the transforming work of the Spirit in ALL believers as mentioned in 11 Corinthians 3:18. To Paul "the ministry" was not something God called him "to do." Rather it was something that was to reveal what God had done in him by His Grace through the Person of Christ.
It was the same for our Lord when He began a three year ministry. It wasn't that He had come to a time and place where He did something "for the Father." Jesus had come to the place where what Had been done in Him was now on public display and it was a ministry that was simply putting before the people what God was about in Him.
"The ministry" is always this only. It isn't, as we seem to think in our present day of professional ministry, a "decision" to go into the ministry. It is, rather, that ALL believers are possessors of the Treasure of Christ that is to be on display in frail human vessels. [11 Corinthians 4:7] Thus, all believers are ministers of the Kingdom news. [Gospel]
The significance of the vessel is never the vessel, whether it be one's age, race or gender. The significance of the vessel is the Treasure there-in.
The significance of preparation for ministry is never the qualifications through training [Seminary] or authority bestowed because of office. [Pastor] It is the significance of what has been done in the vessel [believer] by the Spirit that creates a Christ-consciousness in that vessel, whether man or woman, where He reigns as Lord, and the communication of that reality.
It was the devils who said "Jesus I know." It is THAT which becomes the authority of ministry and especially so when the Body of Christ gathers. We will "know it's Jesus" too. "Who are you?" SHOULD be the question asked by all, not just the devils, when someone tries to minister otherwise.
It is, in fact, the breaking down of the vessel, instead of the building up of the vessel, that is needed for true ministry to be performed which the troubles and difficulties that came to Paul reveal in 11 Corinthians 4:10-12. So a broken and flawed vessel is the place where the Treasure [Jesus] is actually magnified. This, contrary to present day popular opinion, is the true minister, not the one gifted, trained, talented, and capable of presenting truth in a spell binding fashion. It is, rather, the one broken, shattered, and exhibiting the true recovery of Grace that is seen as the true minister scripturally. All believers are to be that minister.
Each christian is that vessel. Each vessel is to reveal that Treasure. Each person is a minister when this is real and is happening. All believers are in "the ministry" though they may provide a living for themselves in many ways. Some may even be supported by a local assembly. But their authority is to only truly come in this way. Not because of an "office" they hold.