Commenting on Paul’s discourse in Ephesus (Acts of Paul IX, 7), Schneemelcher writes in NTA (rev. ed.) 2.218:
Paul further relates in Ephesus that he departed from Damascus – the reasons are not stated, but his departure took place by night, cf. Acts 9:25 – and marched in the direction of Jericho.
This reference to Acts 9.25, however, is not the best explanation of this late departure. It says (RSV, starting at 9.23):
When many days had passed, the Jews plotted to kill him, but their plot became known to Saul. They were watching the gates day and night, to kill him; but his disciples took him by night and let him down over the wall, lowering him in a basket.
Paul leaves Damascus under threat of persecution; while second-century Christians did things at night to avoid persecution, that was not the only reason. Certain events took place at night or at early dawn as a matter of tradition. One such important Christian ritual that took place at the break of day was baptism. Everett Ferguson writes in Encyclopedia of Early Christianity (s.v., baptism, 132):
Second-century sources indicate a period of moral instruction, prayer, and fasting prior to the baptism (Did. 7; Justin, 1 Apol. 61). The Apostolic Tradition of Hippolytus, supplemented by references in Tertullian, provides an account of a developed ceremonial by A. D. 200. After a period of instruction that could last three years, the candidate was examined and prepared for the baptism to occur on the night be Easter Sunday. … Standing in the water, the candidate confessed faith in each person of the Trinity and was immersed three times, once after each confession.
Paul’s departure at night sets up the narrative for the baptism of the lion whom Paul will immerse in water three times (cf. IX, 9) at the dawn following his departure from Damascus (see IX, 7). Later, the Acts of Paul will recount the baptism of Artemilla at dawn (IX, 21). It is unnecessary to assume that the events in Acts 9 have anything to do with Paul’s nighttime departure in the Acts of Paul as the suggestion of Schneemelcher. Indeed, Paul leaves the agape in peace (IX, 7). It is rather the desire to have the lion receive his baptism at the right time of day.