At my dacha, I've run into sudden power outages more than once, and it's always been a real headache. When they cut the electricity without warning for a whole day - that's completely unacceptable, especially in the heat when the fridge is running full blast.
By law, the management company has to notify residents in advance about outages, even if it's scheduled maintenance work. Usually they need to give 3-5 days' notice - it gets posted in the entryway or communicated some other way. If that didn't happen, that's already a violation. And if the outages weren't planned at all and it was an emergency, then the management company is even more responsible for compensating damages, because they caused them.
As for compensation: first, send a written complaint to the management company describing what losses you incurred (spoiled food, damage to the fridge), attach receipts and photos as proof. Make sure to note that the outage happened without proper notice. If the management company refuses - go to the housing inspection authority or Rospotrebnadzor, they can force the company to pay damages. The last resort is taking it to court, but there you'll need to prove direct cause-and-effect between the management company's actions and your losses. Document everything: chat screenshots, receipts, statements from neighbors.