the great thing is that a lot of the best bars are all near each other in downtown SJ which is only really a 7 minute uber ride. I would check out the following:
San Pedro Square. This whole street is really cool and filled with the best bars in the city in my opinion. Besides the San Pedro Market which is awesome itself Ones to check out include:
Old Wagon
Firehouse
All are right next to each other.
Since you mentioned you are near the airport if you like gambling Casino M8trix is a pretty nice casino right across the street. Also has a nice bar.
Food I highly recommend trying when here
La Victoria. The best burritos you will find and people travel for miles just to try their "orange sauce". 2 locations in downtown San Jose including one right next to San Pedro Square.
If you need a breakfast spot I highly recommend Bill's Cafe. Also near the airport.
If you want fancy upscale bars Santana Row is a nice outdoor shopping outlet with tons of bars and fancy restaurants. Nice setting at night.