The first Russian Tsar was Ivan IV, who adopted the title in 1547.
Regarding the buildings:
A fortified residence has existed on the site of the Kremlin since at least the 1090s, though it wasn't called "The Kremlin" until the first half of the 1300s.
The Winter Palace was built during Peter the Great's "westernization" efforts, with construction completed in 1712.
Which means that, by the time the first Tsars came along, the leaders of Russia had been living on the site of the Kremlin for about 400 years, and in a building named that for more than 200. They would keep doing so for another century and a half.