I teach a BJCP study class about once a year or so, so I've heard a lot of feedback on the entrance exam.
You will need to know styles pretty well. Technically its open book but practically you won't have time to look much up.
You'll need at least a rough idea of strength, color, bitterness, malt profile, etc for all of the styles that might appear. That doesn't necessarily mean exact numbers, but you should have an idea of the range that various examples of the style might fall into. The "Impression" and "Comparison" sections of the style guidelines provide concise summaries that are easier to remember than all the detail in the other sections. I'd key on those when prepping for the entrance exam.
The entrance exam will also have some off flavor questions. Know the common off flavors, what causes them, how to prevent them, and when they might be allowable (e.g. a very low amount of DMS in a German pilsner, or very low diacetyl in an ordinary bitter).
There is also some BJCP program info covered in the online exam - know judging procedure, ethics, requirements for rank advancement, etc. Some of this is covered in the Judge Procedure's Manual and some of it is in the study guide.
The tasting exam is a different animal, although you'll still need to know styles in order to provide good feedback and identify where examples fall short. Start prepping for it at least 3-4 months in advance, and preferably longer. The single best thing you can do here is know how to fill out a good scoresheet, which means knowing what the graders are looking for. Read the BJCP Scoresheet Guide, and understand what is meant in terms of perception, descriptive ability, feedback, and completeness. Buy commercial examples of lots of different styles, score them, and then compare your scoresheet against the rubrics in the Scoresheet Guide. Keep in mind that the most common off flavor in store-bought beer is oxidation, which is not necessarily the same off flavor you'll find in homebrewed examples.
The most common flaw I see in scoresheets from new judges is failure to describe every aspect of the beer. This costs you exam points in both completeness and descriptive ability, and maybe perception, depending on what the graders are looking for. Comment on everything in the fine print on each section of the scoresheet, but don't stop there. Comment on everything important to the style, even if it isn't explicitly called out in the fine print - e.g., banana and clove in a weissbier. That's true even if it's not present in the example in front of you. Maybe the admin decided to give you an American wheat and call it a weissbier.
Read the exam admin guide as well. The admin is supposed to provide a range of styles, at least one high-scoring beer, and at least one low-scoring beer. Know what you are getting into ahead of time.
There's a ton of stuff to work on prior to the exam, but at least there's beer involved!