This is becoming one of the most frequently asked questions on this site. Need to add it to the FAQ section.
Yes, you can take a bus from Punta Cana to Santo Domingo or the other way around. Grab the Sitrabapu express bus at the Bavaro bus station. A taxi will probably charge you RD$300 for the 20 minute ride from your hotel.
In Santo Domingo, this air conditioned bus will drop you off at Calle Juan Sanchez Ramirez No. 31, near Avenida Maximo Gomez, Tel. 686-0637. The bus departs from Punta Cana at 7 am and 4 pm. This is a four hour trip that may make a short stopover in San Pedro de Macoris to let off passengers. The cost is RD$120.
From that bus station you can take a RD$60 taxi to your hotel or final destination in the city.
Regarding the scenery... it's changing. You will pass near some sugar fields, along the Caribbean Sea, besides lots of empty land, lots of coconut trees. And of course, there is bustling chaos of Santo Domingo, the capital city.