What kind of account does the Calendar support? I cant add events to the Calendar, nor the default nor aCalendar because I need to have an online account to sync them. 
In Settings/Accounts I have the option to add a Corporate Account (Exchange), Facebook and Yahoo. Facebook and Yahoo do not work, Exchange looks like the right option but I dont have Exchange.
I am using Android 4.1 and it doesn't have the Google applications installed.

Is there another alternative like using a webcal or a local calendar?
Try using the Business Calender app, or one of the offline Calendar apps.