Hi Ursin,
Automatic updates in wordpress mean that the update can be initiated with the wordpress update manager, but it’s not started unless the administrator initiates it. (it’s automatic in a sense that you don’t have to delete and re-install the plugin manually every time similarly to the free plugins)
To do that go to the Dashboard -> Updates menu, hit the Check Again button and Ajax Search Pro should pop-up in the updates list. Now tick the checkbox next to it, and hit Update plugins button: http://i.imgur.com/I0zPy3T.png
