ANSWER: Paul can rightfully be called an apostle because he fulfilled the same three basic requirements the specially-selected twelve original disciples met. After the death of Judas, the church (totaling about 120) met in Jerusalem to decide who among them would be uniquely qualified to replace him (Acts 1).
Generally, those to be considered for the awesome responsibility of an apostle had to have been personally called by Christ, taught by him directly for several years (as Paul would be), and seen him alive after his resurrection (Acts 1:17, 22, 25).
Saul of Tarsus, whose name would later be changed to Paul, was initially a zealous persecutor of those who believed Christ was the Messiah (Acts 9). While traveling to Damascus in 33 A.D., three years after Jesus was resurrected, he was was miraculously confronted by the Lord and brought to repentance (verses 3 to 6). While he stayed in Damascus, a believer named Ananias received a vision that revealed why Paul was called and chosen to ultimately take on a special responsiblity.
But the Lord said to him (Ananias), "Go, for this man (Paul) is a chosen vessel to Me, to bear My name before the Gentiles, and kings, and the children of Israel; For I will show him what great things he must suffer for My name" (Acts 9:15 - 16, HBFV throughout).
After Paul was baptized in Damascus he began to powerful preach that Christ was the Son of God (Acts 9:20 - 22). Death threats from Jews living in the city, who rejected his message, drove him to flee to Arabia. During his three years of exile in Arabia he was personally taught by Jesus (Acts 9:20 - 25, Galatians 1:11 - 12, 15 - 18). At this point in his life he fulfilled the same basic criteria the original disciples did in regard to being considered an apostle.
Additional Biblical evidence exists that shows, and reaffirms, the calling of Paul as being unique. In Acts 13 the church in Syrian Antioch is told by God to "separate both Barnabas and Saul to Me for the work to which I have called them" (Acts 13:2). The church soon lays hands on the two men and sends them out to evangelize (verses 3 - 4). After this event Luke, the writer of the book of Acts, labels both Paul and Barnabas as apostles (Acts 14:14).
Paul himself, in many of his writings, asserts his right to his special calling (1Corinthians 1:1, 2Corinthians 11:5, Galatians 1:1, Colossians 1:1, 1Timothy 1:1, etc.). Defending his unique calling to the churches in the Roman province of Galatia he writes the following.
Paul, an apostle, not sent from men nor made by man, but by Jesus Christ and God the Father . . . But I certify to you, brethren, that the gospel that was preached by me is not according to man; Because neither did I receive it from man, nor was I taught it by man; rather, it was by the revelation of Jesus Christ . . . (Galatians 1:11 - 12).
The word "apostle," used eighty-one times in the New Testament, simply means "a delegate, an ambassador of the gospel, or one that is sent" (Strong's Concordance #G652). Surprisingly, the Bible does not limit the use of this term to Jesus' innermost twelve followers or even to Paul.
As stated previously, Barnabas was also given this designation by Luke (Acts 14:14). Paul, defending himself against those critical of his ministry, asserts to the Corinthians that both he and Barnabas have the same right to be financially supported as the other apostles (1Corinthians 9). Barnabas was considered "one sent" by God to preach the gospel even though (as far as the Biblical record is concerned) Scripture does not state he was personally taught by Christ or saw him after his resurrection.