I live up the road a few blocks from where you took that picture in Ravenna and I ride up the Ave almost every day on my way home from work downtown. I ride a Bianchi Volpe that I converted to single speed. The Volpe is basically the same frame as the San Jose, but set up for gears.
I ride my Bianchi all over Seattle. There aren't too many hills that I can't ride up. Some times I have to stand up and grunt a bit, but eventually I get to where I'm going.
So, yes you can do it single speed or fixed. Personally I prefer SS, but that's me. If you haven't been riding much, it might be some work for you at first, but stick with it and it will get easier. And I love my Volpe, so I think the San Jose would be a great choice. If you plan on riding year round in Seattle, fenders aren't a bad idea and the San Jose will take them much better than a Pista or a lot of other track bikes. Oh and yes, that's a good price.