My understanding is that, since Christ grows out of the Jewish faith, the only thing those in Acts needed were Christ. They understood sin, priesthood, liturgy, etc. All of those baptized immediately in Acts are either Jews or Jewish converts. For example:
-Cornelius (a Roman proselyte)
-Paul (a Pharisee)
-The Ethiopian eunuch (proselyte)
They knew the Jewish faith, they just needed Christ. Their Jewish faith WAS their catechism. However, today, the vast differences of understanding between various sects it is necessary for a period of catechesis. I would argue that pagans in the first century would've experienced the same thing. Keep in mind that in Acts it's recorded that St. Paul would sometimes stay several years with a new mission parish. I would argue that he spent these years catechizing them, and only baptized and ordained priests/bishops soon before he left, after catechesis was complete.
As for St. Constantine, though, it was fairly common practice for people to receive baptism near their deaths so as to avoid sinning too much prior to death, and therefore not horribly soil their baptism. This was practiced for awhile, but eventually an earlier baptism (including the baptism of infants) became the norm again.