We took uber downtown from our hotel (Embassy Suites Napa) which worked great down and back. With your own car, I agree with using the lot behind the bank building housing the restaurant. The parking lot is accessed from Main Street, not on 1st Street.
There's public parking down the block towards Main st
You can park in the parking garage net to the bank. Wells Fargo. That's the best place. But will have to walk a block. Also inside the bank parking lot. Then again you can try your luck by parking in the lot behind the restaurant. .!
Somewhere along Main either side of First your best bet. There are side streets south of First on of Main which might have parking. Likewise there are streets West of Main on First that might have parking.
There's one small parking lot behind the building and another one about a block further back.
There is a garage on Pearl street
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