Sync mail attribute from Entra ID to local Active Direcotry

Synchronization is always one way, from AD to Entra. The only exception is when you have the "Hybrid" setting enabled, which allows a limited set of attributes to be synced back, as detailed here: https://learn.microsoft.com/en-us/entra/identity/hybrid/connect/reference-connect-sync-attributes-synchronized#exchange-hybrid-writeback
"Mail" is not ones of those attributes, but proxyAddresses is, so perhaps you can leverage that.