Does this a common phrase? Or at least does it sound natural?
I know this is quite common:
"Okay, you have the job. How about starting tomorrow?"
But I'm not sure about this:
"Okay, you have the job. How about you start tomorrow?"
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Both your phrases are idiomatic and correct English, however, they are in a very casual register. In some parts of America (such as California), and for certain job types (such as "Blue Collar" jobs), a casual register might be used even in formal situations such as hiring a new employee.
However, many people would consider hiring a new employee to be a situation that requires a formal register, and neither of the phrases in your question would be appropriate. "How about" is simply quite casual.