Both San Francisco and Oakland are a 1 hour drive to Napa, however, you are more at risk of hitting traffic in San Francisco.
Book your stay as soon as possible as hotels are already getting booked up for May 2023.
A great, affordable rental car option is Turo. You can have the car of your choice dropped off at the airport for you when you land. The only risk with this is the possibility of the car owner cancelling on you last minute. Make sure to read reviews before booking.
If you plan on wine tasting the day before or morning of the wedding please make sure you reserve your tasting well ahead of time. From our experience wineries are no longer accepting walk ins.