There was radio and still a fairly-active theater scene with concerts and operas being performed during bombings. Composers often had to fight bitterly with city officials to secure above average rations for their performers to make sure they had the strength to play.

Russia's famous composure Shostakovitch wrote and performed arguably his greatest work, his seventh symphony, during the siege. You can listen to it here, it's supposed to be a thematic representation of the siege.

Sources and further reading would be Leningrad: Siege and Symphony by Brian Moynahan which is a fantastic juxtaposition of the war and the arts that struggled to survive underneath it.