Bank HaPoalim sponsors free activities all around Israel every Pesach. They are continuing the tradition this year as well. Tourist Israel has translated the list of free sites for 2013 into English.
Remember that when visiting these sites it is best to come early (preferably at opening time), before the crowds show up. Some sites become so crowded that they stop letting people in. Try visiting some of the lesser-known and out-of-the-way sites to avoid the crowds.
There are also many free hikes, forests and parks if the weather is nice. Malls often offer free shows and festivals, but remember: a trip to the mall is never free! You always ended up buying something…