If an apostle is the founding leadership gift of a church in an area or in a culture, as I have purported in an earlier post, we could well say that the fledgling church is pastored by an apostle. To be technically accurate, that leader is and should be called an apostle, but he would also be acting as the pastor of that start-up congregation. It seems to me most places today would just call him "pastor."
We could make a case that once a congregation is established it would be more correct to call the leader of that church an elder or bishop, but then Acts 20 indicates that the terms bishop, elder, and pastor are largely interchangeable. Can an apostle even be designated a pastor-- wouldn't that be a confusion of terms since both apostle and pastor are listed in Ephesians 4?
It may seem so at first glance, but a few verses (2 John 1:1; 3 John 1:1; Galatians 1:19; & Acts 21:8) make me think otherwise.
If one takes the Apostle John to be the author of all the Johannine epistles (as I do) and understands James, the brother of Jesus, to be the first Bishop of the Church in Jerusalem, as many do, a biblical case for seeing any of the five-fold leadership gifts as capable of exercising the presiding office begins to emerge. The long and short of it: the office of elder or bishop does not describe the gift that is expressed by the individual occupying it.
Sometimes the elder/bishop will be an apostle (as in the formative stages of the church in a culture or geographic area), sometimes a prophet, or an evangelist, or even a pastor/teacher. I think the same kind of thing is true for the diaconate.
For leadership in the church, it's the function of preparing God's people for works of service so that the body of Christ may be built up rather than the type of gift leading that matters. Any of the five-fold gifts is capable of leading that preparation. It will be the case that some folk in any given church will be gifted in similar ways to the church leader, but that will not automatically qualify them to be church leaders. Church leaders are gifted and full of the Holy Spirit, tested in service, of good reputation with all, and good managers of their own families among other things.
In any church some will be more gifted than others, and if some folks are given to their gifts (i.e. "full-time"), it follows that some folks will not be. But whatever the situation is with any person's gifts we can be assured of this: when a tempered individual has been gifted in the ways leaders are gifted, his gift will make room for him.