I couldn't find any evidence that Putin was hiding in a bunker for a significant amount of time in 2020. As it seems, most of the rumors come from the photos of his apparently windowless conference room in Novo-Ogaryovo.
The earliest reference to the "bunker" I could find was published on d3.ru in 2015: Is Putin in the Bunker?:
Путин в бункере?
Ни разу не видел в новостях такого кабинета. К чему бы это?
Is Putin in the bunker?
I've never seen such a room in the news? What could this possibly mean?
This refers to the conference with the Ministry of Defense on the situation in Syria:
According to official sources, this footage was shot in Putin's residence in Novo-Ogaryovo. They don't go into details as to where this particular room is located, so we can't tell if it's underground or not. The camera angle doesn't show much of a room, so it's possible that it has windows on the wall behind the camera.
These rumors reemerged in March 2020 when Putin announced the nationwide non-working period. The Russian government started to work remotely and Putin was again seen multiple times in the similar windowless conference room:
Again, the official site of the Kremlin sets the location in Novo-Ogaryovo. This didn't stop people on the internet from speculating that Putin is in a bunker hidden somewhere, there's a surge of articles in March: 1, 2, 3.
Putin being in Novo-Ogaryovo is not unusual per se (see 2015 photo), he regularly holds meetings there. The unusual part is seeing him teleconferencing there alone in a giant room. This, together with heightened attention to the government during the pandemic, created a good ground for rumors and jokes.
It's possible that both Peskov (Putin's press secretary) and Navalny are correct. Peskov correctly says that Putin spends his time in his residence and doesn't really need a bunker. At the same time, Navalny uses "a bunker" figuratively to refer to the whole Novo-Ogaryovo estate, surrounded by six-meter-tall walls and isolated from the outer world.