Hamburg has something called Jasper Express which will take you for 5 Euros directly without stop to Hamburg train mainstation (8 Euro return ticket). You can go through the hassle of doing the longer but cheaper trip but in the end you have to go for your preference time over money.

The airport express basically leaves around every 15 min, they do have a website in english
http://www.jasper-hamburg.de/index.php?id=6&L=1

There is to be build a direct connection (finally ...) but I think it opens up like 2008 or so.

From there it is a short stroll into the city and shopping street with some nice cafes and such. Near the Airport is nothing to write home about. You do have a little bit of shopping possiblity inside of the airport