connect a wireless card to arduino

I have this wireless card: http://www.bargainstock.co.uk/Photos/Belkin_F5D7011ef.jpg

can I connect this to arduino rather than buying a regular wifi shield?

Sorry no.

Cheapest solution would probably be to get a cheap router that can run OpenWRT or Tomato Linux and connect the serial lines to the Arduino. There’s also the option of building your own ENC28J60 circuit.