Do they automatically update their search index or do i have to do
something to get it updated? ... Plus if it is auto, how long does it
usually take?
Historically, google was using their home-brewed mapReduce functionality to rebuild their search engine every few days, which caused this type of behavior. It was / is automatic, meaning you dont have to do anything special to cause google to update their index of your site, in particular.
However, according to a new wired article about their "Colossus" product, they're no longer doing that intervalled mapReduce, and now are favoring a more real-time index rebuilding algorithm.
With that said, the jury is still out on how quickly one can reliably expect their site to be re-indexed. You may just wait and see, and take a benchmark about it once google does update your content. Then publish an article about it... you'll be famous. ;)