The answer to what kind of coach he will be should be somewhat obvious from his past - straight-up, aggressive (at least with his defensive schemes), not brash and bold like Rex and is considered a players coach (for different reasons than Rex).
The real question is will he be successful.
He like many other coordinators turned HC has had great success in managing one side of the ball (it's the reason they got hired). It's such a quantum jump in taking on the whole team that many have failed. Some have needed time and experience (like Pete Carroll) and some/most never make it.
What we need is someone who will manage the entire team effectively, work well with the GM in getting the right talent for the schemes they run, pick a good staff, develop talent and constantly get better or learn from your past mistakes (something Rex didn't do). Hopefully, Bowles will bring this to our Jets.