Good Question. When I write "the chicken bus". That's just what people call it. It really means all public buses in the country. All the buses minus expensive tour buses are chicken buses. Which are basically old United States School Buses as you can see in the photo.
So from Costa Rica you would cross the border by land. Then take a "chicken bus" to El Virgen....name of a town. Then change to another Chicken bus direct to San Juan del Sur. If you have some Spanish ability it would be easier. Likely you can find a friendly local passenger able to help, a middle aged women is a safe bet.
From the border of Costa Rica, you will get endlessly hassled to take a private shuttle for between 20 and 40 dollars. If you can split the price with some it may be worth it and quicker. But you'll miss out on the adventurous "Chicken Bus Experience" with the locals. The whole trip from the border of Costa Rica costs like 1.20 USD.
The reason they call it a chicken bus is because you might be sitting next to a local who is bringing dinner home....a live chicken!
Hope this helps!