Most meteorologists do shift-work for some part of their career. I did. People want an up-to-date weather forecasts at all hours, and the weather doesn't go to bed. So some meteorologists have to be up to prepare it. Just as with nurses and doctors, it's part of being in a job where you help people. I found the interest of the job more than compensated for the shift work. Many meteorologists like the weather forecasting challenge so much they spend their careers doing shift-work.
In a sense meteorologists are almost always wrong. If you forecast the temperature to be very slightly too low (or high) some would say the forecast is wrong. That type of person is called a perfectionist. They usually lead a miserable life because for most problems in life you can rarely have the exactly right answer. But you can be close enough to being exactly right to be helpful to people. And providing helpful information is what most meteorologists care about. Even with all the weather satellites, radars and supercomputers sometimes you make a mistake, and you do feel bad about it. If you did your best then its not your fault; the science of meteorology isn't perfect. You just have to tolerate the complaints, knowing that most times you are accurate enough to be helpful. In summary on your two good questions. You have to be physically tough to stand shift work, and mentally tough to be able to find satisfaction in a job where errors (mostly small) happen all the time, and large errors happen occasionally. Some people are not tough enough, and it's good to think about whether you are before deciding you want to be a meteorologist.