There is considerable difficulty in harmonising the chronology of Paul's epistles with that of Acts, and there is no scholarly consensus that Acts 9:26 accurately describes the same event as Galatians 1:18. There is also the issue of whether the author known to us as Luke intentionally omitted secondary details from Paul's epistles, with the result that chronologies will never line up.
Luke's sources
Coleman A Baker (Identity, Memory, and Narrative in Early Christianity) says (page 35) the date of composition of Luke-Acts, which he assumes to be the last decade of the first century, is not the same as the time period narrated in the text. So, either Luke was writing from oral traditions that had come down to him, or he had access to Paul's Epistle to the Galatians (and other epistles).
Baker says (page 66) it is widely acknowledged that Acts and Paul's letters do not agree on the chronology of Paul's life and career. For that reason, many scholars have maintained that Luke did not have access to Paul's epistles when Acts was composed. On the other hand, Heikka Leppӓ has put forward linguistic evidence, primarily similarities in vocabulary, that Luke had access to Galatians and, in writing his second volume, sought to subvert Paul's account.
If Luke was working from oral traditions about Paul it ought to be surprising how similar his account is to that of Paul, in spite of the many differences, and we could assume that the Act's chronologies would be the same as that of Galatians if only Luke had better sources.
If Luke was working from a copy of Galatians, he may have compressed the narrative from Paul's epistle so as to omit part of the three years of which Paul speaks. Craig S. Keener ([Acts: An Exegetical Commentary : Volume 2) tells us such compression or omission was acceptable historical and rhetorical practice. On this view, Acts would be historically and chronologically accurate, although there is the appearance of chronological misalignment.
Historical accuracy
Sir William Ramsay (1851-1939) stated, "... the book could bear the most minute scrutiny as an authority for the facts of the Aegean world, and that it was written with such judgment, skill, art and perception of truth as to be a model of historical statement." In contrast, a view held by many scholars today is that Acts of the Apostles is only loosely based on history, in which case it may not be useful to attempt to match its account with that in Paul's epistles. Uta Ranke-Heinemann says in Putting Away Childish Things that anyone who has kept up with New Testament studies over the last century can only laugh when they hear views like those of Ramsay. John Dominic Crossan says in The Birth of Christianity, page 21, Acts of the Apostles is theology rather than history. Which to believe - Ramsay's view that Acts is an accurate historical document, or modern scholars who say that it is to a large extent not historical?
Acts 9:23-25 says that the Jews sought to kill Paul and that the disciples lowered him down the wall in a basket. 2 Corinthians 11:23 has a very similar account of Paul being lowered down the wall in a basket, but this time it was not the Jews but the pagan garrison, acting under instructions from the city governor, who sought to apprehend him. Acts chapter 9 has reworked Paul's historical account for theological purposes and it therefore need not correspond literally with Galatians.
Conclusion
From his own account, Paul's escape from the governor appears likely to have been at the end of a second visit to Damascus, but it is possible to place this at the end of a first visit. However, when Acts 9:23 refers to many days, it seems unlikely that this was a reference to a period as long as three years, else Barnabas could scarcely have needed to explain to the apostles in Jerusalem that Paul had seen the Lord and that Paul had preached boldly in Damascus. At the end of three years, these things would have been common knowledge.
