Both look okay to me. A better way to say it might be "She made up her mind to put makeup on" or an even more concise statement, "She decided to put on makeup."

It depends on what you mean. If you mean “put lipstick, eye-liner, etc.” on her face, then “put her make up on” is right. “Made up her face” suggests that, after making up her mind, she decided what EXPRESSION to show on her face (e.g., a polite smile, a determined look, etc.)