The Sunday obligation applies to the modern Sunday, reckoned from midnight to midnight. This was established by canon 1246 of the 1917 Code of Canon Law.
The ancient Jews reckoned days from sundown to sundown, meaning that for them the first part of the day was evening. This is why Genesis 1 says things like, "And there was evening, and there was morning--the first day" (Gn 1:5). The same custom was observed by the ancient Phoenicians, Athenians, Arabs, Germans, and Gauls. Today Jews and other groups who keep the sabbath, such as the Seventh-day Adventists, continue to celebrate it from sundown Friday to sundown Saturday. This way of reckoning time was not the only one in the ancient world. For example, the Romans reckoned days from midnight to midnight--the system we use today.
The option of attending an anticipatory Mass on Saturday evening has nothing to do with the fact the sabbath began at sundown. This provision was originally introduced for Catholics who had to miss Sunday Mass for a good reason (for example, because they had to work). The 1983 Code of Canon Law simply states: "The precept of participating in the Mass is satisfied by assistance at a Mass which is celebrated anywhere in a Catholic rite either on the holy day or on the evening of the preceding day" (can. 1248, 1).
Sunday is often spoken of as "the Christian sabbath," but this is not a technical description. Sunday is not a strict replacement for the sabbath (which has been abolished), but a day the Church instituted to fulfill a parallel function. Thus Ignatius of Antioch, the earliest Church Father to address this question, states that Christian converts "have given up keeping the sabbath and now order their lives by the Lord's Day instead, the day when life first dawned for us, thanks to him [Christ] and his death" (Letter to the Magnesians 9 [A.D. 107]).