The League of Legends launcher uses server aliases that have problems connecting to some ISP default DNS servers. If your launcher never seems to start downloading a patch, you may want to change to public DNS servers in your network settings. Most of the time, this only applies to players outside of the US.
WARNING: Changing these settings may cause problems with your internet connection if you have specified network configurations for another purpose. Be prepared to revert to your old settings. Save any information you are about to change.
Changing to Public DNS
Open System Preferences
Open the Network preference pane
Select the active connection on the left, and click the "Advanced..." button in the lower right hand corner
Click "DNS" from the selection bar.
Remove any current DNS entries, and input the following public DNS servers: 18.104.22.168 22.214.171.124
The DNS Servers field should appear as follows:
Save your changes by selecting “OK”
Apply the changes by clicking "Apply" in the lower right hand corner