I didn't start practising until a couple of weeks before I sat the LNAT in 2013 and my score got me in to Oxford. Past papers are the best you can do to prepare, but don't bog yourself down in them- this isn't something you can be completely ready for. This is something designed to assess you as you are, unlike exams which test what you know.
Essay technique and logical thinking are things to perhaps start thinking about during the summer before I'd suggest. But even then you'll never feel prepared for it- no one does!. Remember the average every year is only about half- just keep a level head, an eye on time, and get used to the structure and you'll be fine.
Good luck!