Iceland wasnt cheap when I went.  It was 2008 so pre crisis.  If I remember well you have red roads, these are cool and normal roads, you then have brown roads which are fairly close to off roading, then there are dotted brown roads, we never made it on to these!

I had 3 punctures there in a 4x4.  to tackle anything more than red roads you need the balloon tires.  def check pressure correct and you can undo the wheel nuts before leaving the city, needless to say be savy on changing a tire.

Not to put you off.  Iceland is amazing and its a great trip.  very different landscape and just incredable.  the blown tires are all part of the storys and tales to tell when you get back.  everyone does a driving trip there so you wnt be alone.

If you want more info feel free to contact me. 

Wow, three punctures ?! Good to know that the car rental offices offers 24/7 service. Thanks for you advices.