Plus I showed up with a coat fresher than wet paint
A coat is a painting term; basically it means another layer. So if the paint is still wet after painting a car, house, etc. that means that the coat is still fresh. But Em is saying his coat (as in jacket), is as fresh (as in stylish) as an actual fresh coat of paint.
In addition, fresh (wet) paint has gives off strong fumes. So smelling it can make you faint, referring to the previous line.
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