The main tube will fill up to above the tube leading to cup 3....

The main tube will fill up to above the tube leading to cup 3. As the opening is on the same side as the tube leading to the cup, the tea will run down the outside of the tube and then into the cup 3. The tea will also run down the main tube hitting and then running down the tubes to cup 6, cup 4 and cup 5.
Because the holes on the main tube are smaller than the diameter of the main tube, the main tube will continue to fill past the tube leading to cup 3 up the main tube to the tube leading to cup 7 and maybe onto the tube leading to cup 1.

Therefore the cups will fill in the following order: 3, 6, 4, 5, 7 and 1 and maybe some splash into cup 2.