What is the best way to prepare for this admissions test?

Practice, practice and more practice.

For the language assessments, it is best to do plenty of grammar exercises and read as much as possible in the given language.

For the 'invented language' section for those applying for a beginners language course, it is best to find similar exercises, of which there are plenty online. This part of the admissions test is essentially a sophisticated puzzle. Practising getting to grips with unkown languages can be achieved through mock tests online. I also recommend exploring various different puzzles in order to stretch your brain as much as possible. One must train oneself in highlighting patterns.