Is the Lord's Sabbath on a Saturday or Sunday? What hour does it begin?
The Sabbath is the seventh day of the week, which is currently called Saturday in English speaking countries. The Sabbath was designated as the day of worship for the Israelites living under the Old Law.
"The LORD our God made a covenant with us in Horeb. The LORD did not make this covenant with our fathers, but with us, those who are here today, all of us who are alive. The LORD talked with you face to face on the mountain from the midst of the fire. I stood between the LORD and you at that time, to declare to you the word of the LORD; for you were afraid because of the fire, and you did not go up the mountain. He said: ... Observe the Sabbath day, to keep it holy, as the LORD your God commanded you. Six days you shall labor and do all your work, but the seventh day is the Sabbath of the LORD your God. In it you shall do no work: you, nor your son, nor your daughter, nor your male servant, nor your female servant, nor your ox, nor your donkey, nor any of your cattle, nor your stranger who is within your gates, that your male servant and your female servant may rest as well as you. And remember that you were a slave in the land of Egypt, and the LORD your God brought you out from there by a mighty hand and by an outstretched arm; therefore the LORD your God commanded you to keep the Sabbath day" (Deuteronomy 5:2-5,12-15).
In Israel, days were measured from sundown to sundown. This is because when God created the world, He said evening and morning were a day. So the Jews continued to mark days by starting with the evening.
This covenant that was only for the Israelites, was later to end.
"Behold, the days are coming, says the LORD, when I will make a new covenant with the house of Israel and with the house of Judah- not according to the covenant that I made with their fathers in the day that I took them by the hand to lead them out of the land of Egypt, My covenant which they broke, though I was a husband to them, says the LORD. But this is the covenant that I will make with the house of Israel after those days, says the LORD: I will put My law in their minds, and write it on their hearts; and I will be their God, and they shall be My people. No more shall every man teach his neighbor, and every man his brother, saying, 'Know the LORD,' for they all shall know Me, from the least of them to the greatest of them, says the LORD. For I will forgive their iniquity, and their sin I will remember no more" (Jeremiah 31:31-34).
The new covenant would be different than the first covenant. "In that He says, "A new covenant," He has made the first obsolete. Now what is becoming obsolete and growing old is ready to vanish away" (Hebrews 8:13). From the time of God's declaration that there would be a new covenant in Jeremiah's day, the current covenant became old and obsolete.
"Therefore remember that you, once Gentiles in the flesh -- who are called Uncircumcision by what is called the Circumcision made in the flesh by hands -- that at that time you were without Christ, being aliens from the commonwealth of Israel and strangers from the covenants of promise, having no hope and without God in the world. But now in Christ Jesus you who once were far off have been brought near by the blood of Christ. For He Himself is our peace, who has made both one, and has broken down the middle wall of separation, having abolished in His flesh the enmity, that is, the law of commandments contained in ordinances, so as to create in Himself one new man from the two, thus making peace, and that He might reconcile them both to God in one body through the cross, thereby putting to death the enmity. And He came and preached peace to you who were afar off and to those who were near. For through Him we both have access by one Spirit to the Father. Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God, having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone" (Ephesians 2:11-20).
Jesus is the one who actually changed the law by removing the old to bring in a new. That change was essential for the salvation of the Gentiles since the old law given to Moses was limited to the Israelites. The new covenant was for Jews and Gentiles alike.
"In Him you were also circumcised with the circumcision made without hands, by putting off the body of the sins of the flesh, by the circumcision of Christ, buried with Him in baptism, in which you also were raised with Him through faith in the working of God, who raised Him from the dead. And you, being dead in your trespasses and the uncircumcision of your flesh, He has made alive together with Him, having forgiven you all trespasses, having wiped out the handwriting of requirements that was against us, which was contrary to us. And He has taken it out of the way, having nailed it to the cross. Having disarmed principalities and powers, He made a public spectacle of them, triumphing over them in it. So let no one judge you in food or in drink, or regarding a festival or a new moon or sabbaths, which are a shadow of things to come, but the substance is of Christ" (Colossians 2:11-17).
Those who came from the Gentiles to become Christians were not expected to keep the regulations regarding food, the festivals, or the sabbaths. These were a part of the old covenant which was merely a shadow of the new covenant. An example is given of entrance into the covenant. Under the old covenant it was done through circumcision; that was replaced with baptism -- a spiritual removal of sin instead of a physical removal of skin.
To try and mix the old with the new would not work.
"Stand fast therefore in the liberty by which Christ has made us free, and do not be entangled again with a yoke of bondage. Indeed I, Paul, say to you that if you become circumcised, Christ will profit you nothing. And I testify again to every man who becomes circumcised that he is a debtor to keep the whole law. You have become estranged from Christ, you who attempt to be justified by law; you have fallen from grace" (Galatians 5:1-4).
Under the law of Christ (Galatians 6:2), the day of worship was set to be the first day of the week (Acts 20:7; I Corinthians 16:1-2). That day is significant to Christians because it is the day on which the Lord arose from the grave. It is also the day when the church was established. What was done for worship also changed. Under the Old Law, there were prescribed sacrifices and everyone was required not to work. Under the New Law, we gather for worship and partake of a memorial meal to remember the Lord's death (I Corinthians 11:23-26).
When the day starts used to depends on what country you lived in. Today we all mark days as starting and ending at midnight, as the ancient Romans did. The important point is that Christian worship is done on the first day of the week, the hour chosen for worship is left up to each congregation, so long as it is within the first day of the week.