That $150 taxi fare that you quote is to Bahia Principe La Romana. That place is much farther from PUJ than the town of La Romana. Bahia Principe is the location of the Nueva Romana development that we have talked about in other threads here. It almost all the way to San Pedro de Macoris.
The going rate from Punta Cana to La Romana is $100. At the airport you can tell them where you are going and you should be able to get much less than $150. But look, here is an example of igoring good advice from people that know what they are talking about. See #2, #4, and #5 above. You are at PUJ. The incovenience and cost of doing anything other than going directly to La Romana in the taxi (unless it is night) doesn't make sense.
If you went somewhere else from PUJ, like to find the gua gua to La Romana at Coco Loco or Friusa, that will cost you time and money.
In the taxi, you will get to know the taxi driver. With your fluent Spanish. This is worth something.