The booster isn't because the first shot isn't working now. It will be required because your body will eventually forget how to fight COVID (you'll need three shots no matter when you start) or because new variants will pop up. Either way, your choice is "get the shots now and be safe for the intervening year, or get the two shots a year of non-protection later". The cost of getting another shot every year is nothing. I'd much rather get a shot every year than go a year without being protected.

Why should I study now instead of revising before the exam?

Many reasons. To minimize the chances our health care system is overrun and new mutations don’t emerge it’s beneficial to maximize public immunity to as many variants as we can as early as possible. Additionally the booster shot may come in the form of a third dose anyway, so there may be no personal benefit to waiting either.