Everyone experiences emotions. However some of us cope with and manage our emotions better than others. This is not an innate ability. All humans have the capacity to self-regulate their emotions. But like any other skill, it takes practice and requires experienced teachers who can model and instruct on the way to do this.
Meditation and mindfulness is at its core emotional regulation. It's recognizing and allowing all emotions that come up, and then making conscious choices about what action to take—if any—based on the information that emotion is conveying.
Buddhist monks are masters at this. It might appear that they do not experience the same strong emotions as others, but they do. In fact there's a lot of evidence that they experience their emotions more intensely and fully than most of us. The key difference is they have a mastery of all the skills necessary to manage those emotions, process them, and let them go.
A key concept in all of this: we are all each responsible for managing our own emotions. No one else can do it for us. And similarly, we cannot do it for anyone else but ourselves.
So as you practice mindfulness, keep in the back of your mind that it is part of the process for you to become more comfortable with the emotions that come up for you and how you respond to them.