This year, we have the unusual situation of the first day of Succot, falling on Shabbat. After the destruction of the Second Temple, our sages decreed that we do not take the Lulav on Shabbat, for fear of carrying the Arba Minim from one domain to another.
When the Temple was in existence, they did take their Lulav and Etrog, even when the first day of the holiday, fell on Shabbat. This was in spite of the fear of carrying on the Chag.
They also needed to devise special leniencies for the bringing of water from the Shiloach, for ניסוך המים, water drawing, as well as bringing the ערבה, the willow, for its special ritual.
In the case of the Lulav, all were advised to bring their Lulavs before Shabbat, and place them on special shelves in the Temple. In the event that someone would take the wrong Lulav, a declaration was made. “Anyone whose Lulav should mistakenly come into their hands, should view it as a gift, and is permitted to use it.” This solved the problem of carrying on Shabbat, and the declaration solved the problem of taking the wrong Arba Minim.
We are told that when the third Temple will be rebuilt, we will go back to the old system, and again bless our Lulav and Etrog on Shabbat. Hopefully, that time will come soon!