Well if its rated as !A, then you can use it to draw 1A for long time, its just that your IC will get heated up, and after come time it will reach a temperature equilibrium. If you want to draw 1A for long time, you need to consider all the testing parameters with which IC was tested for rating as 1A. Rule of thumb is provide a proper heat sink and cooling to your IC and you can have rated performace.
If you want to use fuse for prevention, i would say you need a very less current fuse. For instance lets do some calculations. on output you are getting 5V and 1A, so that comes out to be 5W power output. Generally these regulators have an efficiency of 85-90%,( you can check the datasheet for exact value), but lets say efficiency is 85%. so you have 0.85*Vi*Ii = Vo*Io = 5W. so Ii = 5/(12*.85)≈ 0.5 A. So it will be a good idea to put a fuse of rating 0.8-1.2A. Its a good idea to put a fuse of rating 2-3 times the nominal current.