I've been brought up to believe the apostles were a special breed of disciples, and that Peter's "job description" for apostleship as laid out in Acts 1 seems to suggest that the number one criterion for being an apostle is the following (in bold print):
"'Therefore it is necessary that of the men who have accompanied us all the time that the Lord Jesus went in and out among us--beginning with the baptism of John until the day that He was taken up from us--one of these must become a witness with us of His resurrection.' So they put forward two men . . . and the lot fell to Matthias; and he was added to the eleven apostles" (Acts 1:21-26, excerpts).
Significantly, Matthias "was added to the eleven apostles" (v.26) following Judas's demise, to make the number of apostles twelve in number once again. That the number of apostles and the number of the tribes of Israel is 12 is also not a coincidence, I suggest.
Be that as it may, where does that leave Paul? In his own words:
"and last of all, as to one untimely born, [Jesus] appeared to me also" (1 Corinthians 15:8 NASB).
According to the NET Bible Notes, "The Greek word used here (ἔκτρωμα, ektrwma) refers to a premature birth, a miscarriage, or an aborted child. Paul uses it as a powerful figure of the unexpected, abnormal nature of his apostolic call."
When Paul speaks of his "untimely birth" as an apostle, then, he is saying in effect,
"I am not worthy even to be in the company of the Twelve, since I was a persecutor of the church when the 'real apostles' were just getting started in their world-wide mission. I even held the garments of the men who stoned the Apostle Stephen to death! So apart from God's grace you would be justified in saying I was an abnormally-born, bottom-of-the-barrel apostle!"
Why God chose Paul to "sneak under the wire," so to speak, I do not know--other than to say He did so for His own sovereign purposes (see Galatians 1:15). Some Christians, I suggest, have gone out on a limb by saying the eleven's selection of Matthias by lots was a mistake, and that Paul, not Matthias, should have been the twelfth apostle, if the eleven had only waited on the Lord! I do not give this argument any credence, however, and I chalk up Paul's being "unlucky 13" to God's election of him as number 13!
As for whether we have modern-day apostles, I think the answer is clearly no! I am not saying Jesus could not (and has not) appeared to people since the time He appeared to Paul on the road to Damascus. I am saying God in his wisdom wanted there to be only 12 apostles and one mutt, a "special case," the runt of the litter, the Apostle Paul! Thank God for him, too!