does the game auto-update or do I need to manually download any updates?
The beta has an auto-update capability.
From cliffski in the Support Forum:
"Ok, the game autoO-checks for updates once a day, but if you just missed it, you can either…
re-use your original download link from BMT micro and grab a fresh clean installer, which is now version 1.16
or
Delete the file called ‘up’ in \my documents\my games\gratuitous space battles\web
Then run the game."
As a quick addendum:
If deleting the up file does not work, open it in text editor and change the date to yesterday.
( when I deleted the up file for some reason it did not download the patch 
)