Just after leaving the airplane, there's passport control, which is long because they're understaffed.
It takes 45-60 minutes to get to taksim square, by either shuttle bus or metro. The bus because is easier, the metro is more dependable. A bus leaves every half hour. Hop in, the driver will walk and ask all passengers for money. In Turkey euros and dollars are commonly accepted.
Taksim square is pretty boring for a tourist. The nicest area is between Sultanahmet and Eminonu.
Your plan is totally doable.
EDIT: The bus has "Havatas" written in large letters on the sides, you can't miss it. It costs 14 lirasi, around 5 euros. It's a direct bus to Taksim, if i recall correctly (in all other turkish cities i visited, it was not direct).
Alternatively, for the metro, you need to take the red line to the terminus, Aksaray, and from there take the tramway direction Kabatas, and it's 4 stops to Sultanahmet. There you'll find Hagia Sophia and delicious ice cream.