Hi L2C, don't worry too much with any European city with regards to Sunday opening - especially if they rely on tourism & you visit in the peak of Summer, nearly everything will be open & very lively.
If you're not sure what to do in Villefranche, may I suggest taking a look at http://www.cruisecritic.com/ports/newport.cfm?ID=84
which seems to sum everything up quite nicely, with photos as well.
Whilst on the French Riviera, it is also quite easy to use the train service to visit other towns such as Nice & Cannes, or even another "country" with Monte Carlo/Monaco along the coast.
Hope you have a lovely time,
Alan & Katrina
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