I don't think that the fear here is another meltdown but just general release of radioactive waste in general.
According to the New Scientist article Chernobyl power cut sparks fears of potential for radiation leaks, it seems like the issue here is radioactive waste from when it was in opeartion.
Fears of a potential radiation leak at the Chernobyl nuclear power plant are growing after a power cut at the site. Without electricity it will be difficult to cool ponds that contain hazardous nuclear waste and to filter the air inside the vast containment building that houses the remains of the reactor that was destroyed during an infamous disaster at the site in 1986.
It seems the concern is keeping the spent fuel properly cooled in order to keep it covered with water to prevent the spread of radioactive material and to keep the filters operating that help contain it.
Spent nuclear fuel from Chernobyl’s former reactors is stored in a large cooling pond that is constantly replenished with fresh, cold water to keep its temperature down. Without an electricity supply – which the Ukrainian government says the site now lacks – this cooling has stopped, which will allow the water temperature to rise and increase the rate of evaporation.
If the ponds are allowed to run dry then there is a chance that radioactive material could be released into the environment. There are also concerns that electrical air filters in the containment building around the reactor will stop, causing condensation that may harm the building.