Hello Roula,
The Museo Archeologico al Teatro Romano appears to be closed on Mondays. The Arena opens later on Mondays but is still open - pending opera and other events that might be taking place. The church in San Zeno is open.
These are the sites I would care about but I guess it depends on what you wanted to see in Verona and you'll have to check on those. I consider the old city itself to be one big attraction to wander through and so that never closes.
I'm sorry I don't have a specific restaurant suggestion that would be open early.
Verona is great, enjoy it,
=Tod