The drive from Santiago to Punta Cana should take you between 5 to 6 hours. Just fron Santiago to Santo Domingo is around 1 1/2 hours to 1 hour and 45 minutes, and then just to get across the city count like 30 minutes until you hit Juan Bosch Bridge to take Las Amerixas Highway. Then you will pass by SDQ and hit the new Autovia del Este (which is not done yet). Then, you can get to La Romana like 1 hour and 15 minutes. After that, you have to go and pass the city, pass Casa de Campo and take the autopista that connect La Romana with Higuey, which is only 2 lines (one each way). Then you will pass inside Higuey, pass by la Basilica and make couple of turns until you find the sign for Punta Cana-Bavaro, which is also a 2 lines autopista. If you make it in less than 5 hours you are a F1 driver.
Good Luck... But let me tell you, it is a nice ride to see a lot of stuff. When you hit La Romana, you can stop in La Represa, Altos de Chavon, etc.