At least three hours, via San Francisco de Macoris, or down the Duarte highway until Piedra Blanca and then turn right via Maimon. Both these routes may have been affected in last year's tropical storms so best to check which is in better condition now.
the last time i went to cotui it was left off the duarte highway....just before the hospital, but if you go via the san fransisco de macoris way then it is a right....about 3hrs though and a lot of the highway has been repaved
Thandie: It can take anywhere from about 3.5 hours to more than 5 hours depending on the route you take. If you take Duarte Highway (shortest route), past Santiago and La vega, turn left at "El Pino" towards Rincon, just past La Vega. In Rincon turn right at the police station (If you continue to go straight then you'll go all the way to San Francisco de Macoris which is about an hour longer drive).
You can also go all the way to Piedra Blanca as Chiri suggested, but that's about 45 minutes longer.
The roads into Cotui are all really bumpy nowdays as they have washed out, etc and they haven't been fixed yet...and BTW do not go through Pimentel as the bridge was washed out about two years ago and it still hasn't been fixed.
If you have plenty of time to kill I suggest you take the scenic route, west of Puerto Plata towards Cabarete and turn towards Jamao, Moca, etc...This is a beautiful drive by the sea first and then through the mountains ending up in San Francisco. Much longer drive though.
PM me if you need info about Cotui (I'm from there)