How much time you need varies a lot from person to person. What can be said with confidence is 'which order gives the highest return.'
Honestly, no reliable number can be given. It varies by multiples depending on your native language, learning environment, and hours per week.
It's generally understood that living in Korea and studying intensively differs a lot from studying a few hours a week abroad. But that's only a reference point, not a guarantee.
A more reliable indicator than time spent is a past paper. Solving one past exam under real timed conditions tells you exactly where you stand right now. How many points remain to your goal is the real starting point of a plan.
By return on effort, the order is Writing Q51·52 → the early sections of listening and reading → Writing Q53.
Q51·52 are worth 20 points for just two sentences. Since you only need to learn fixed sentence patterns, they give the highest output per unit of effort in the whole exam.
The early sections of listening and reading are low difficulty, so raising your accuracy there is easy, yet each is worth the same 2 points as the harder questions later. It's always a better deal than getting stuck on a hard question.
Analyzing one paper properly beats solving many. If you just grade it and move on, you'll miss the same question type again next time.
For every wrong answer, write down 'why I picked that choice.' The fix is completely different depending on whether you didn't know the vocabulary, missed the grammar, or simply ran out of time.
Solve at least one paper start to finish under real timed conditions (60 minutes listening / 50 minutes writing / 70 minutes reading). The score gap between having time pressure and not having it is large.
This site's study-method card, 'Why doesn't your score improve if you only grade past papers?,' covers the analysis method.
Listening, reading, and Writing Q51–53 can be fully prepared through self-study. The answers are fixed, so you can grade and correct yourself.
The hard part is Writing Q54. An essay is difficult to grade yourself, so it needs outside feedback. But if Level 3 is your goal, the strategy of just filling the length and format requirement for Q54 still works, so self-study's limits aren't a big issue there.
Speaking isn't part of the exam, but if real conversation ability is your goal, you need a conversation partner. TOPIK score and speaking ability should be treated as separate things.
Spend your time practicing retrieval of what you already know, rather than learning anything new.
The Writing Q51·52 patterns and the Q53 template still convert directly into points even right before the exam. Write both out by hand, repeatedly.
Write your time-budget plan on paper and memorize it — for example, 10 minutes for Reading Q1–12, 15 minutes for Q13–24. The fewer decisions you have to make in the exam room, the fewer mistakes you make.
Practice listening at the same time of day as the actual exam. If it's a morning exam, listening to audio in the morning tunes your ear to that time slot.