No, you are not allowed to use the JavaDoc (or any other reference material.) Anything you need to know needs to be in your head.
If you dont' already have a study guide, I recommend buying one. That distills the information you need to learn. Granted it is still a lot. But at least you won't waste time and mental space on APIs known not to be on the exam.
What Jeanne says is true. It is a lot. I am also wondering if I will ever be able to get that knowledge into my head.
I am currently preparing for the OCP exam using this OCP study guide, which I highly recommend. It clearly mentions what one must know as per the exam objectives and also gives plenty of context, example code snippets and explanations.
Beginning programmer - self-taught
Oracle Certified Associate Java SE 8