I had the same problem. I found that after the patch is downloaded Android Studio tries to update itself but as it is run without sudo rights it fallbacks to simply run the old version of Android Studio without updating it.
In order to update my old version of Android studio I went to the folder where Android studio was installed, in my case ~/apps/android-studio/bin/, and run with sudo Android Studio, in my case sudo ./studio.sh
It will then ask you for your Android Studio configuration file. Be careful about the location where the installation process will install a new Android SDK version, because in my case it wanted to install it in the wrong directory, I had to select custom installation in order to select the folder where I had my previous Android SDK.