An apostle is one called by God to establish the church where it had not been established before. The pattern for this ministry was set by the twelve Apostles of Christ, who defined by words and deeds what an apostle is and would do in the time after them. The only real distinguishment between the twelve and the apostles who followed is that the twelve were hand-picked eyewitnesses of Christ's resurrection and the revelatory source for what Jesus said and did, whereas the apostles who followed were not. The apostles who followed rely upon the witness of the twelve but cannot add to it.
Much is made of apostolic authority today, whether focused on the twelve or the latter variety, but I see no record in the scriptures that apostolic authority was ever derived from anything other than anointing, ethos, consensus, and ultimately persuasion. That kind of authority is acknowledged by those under it, but cannot be claimed hierarchically, dictatorially, or oppressively. An apostle who insists upon that kind of authority would be establishing a cult not the church.
I think it self-evident that an apostle would be a generalist in terms of gifted skills. He is the church when he starts. All that needs to be inspired is going to have to be inspired through him, hence he will be a jack of all trades. Then, through preaching, teaching, and signs and wonders, others will be added to him birthing a growing church. As it grows, believers will mature and others will step into specific aspects of gifting for which the apostle was a generalist.
So, as the body grows prophets will arise, as will teachers, and all manner of other utilitarian gifts as Paul taught in 1 Corinthians 12:28 (or perhaps as implied by the listing in Ephesians 4:11). The ordinals used in the Corinthian passage are not a reflection of rank or authority, but of the order in which ministry gifts arise in the extension of the body as it grows. Apostles begin things that are then carried forward by others; therefore, for a mature church to "look backward" toward reestablishing an "apostle" is regression, twisted logic and just bad practice.