Shared shuttles operate on a fixed schedule, once or twice a day. There are more options for a direct bus but you’d need to head into downtown San Jose to Terminal de Autobus Atlantico Norte located on Calle 12 between Avenidas 7 and 9 to catch the Mepe bus. Lots more detail & schedules at https:/…transport-to-puerto-viejo.php
Assuming you mean SJO, not Liberia international airport:
We used PleasureRideCR shared shuttle (they pick up right from the SJO terminal once daily, but need at least 2 ppl per ride, so if you are traveling solo and they have others on that date, they can add you, otherwise you pay for 2 tickets). We also used Interbus shared shuttle vans in between S. Caribbean and SJO. Those have a morning and and an afternoon ride and had more ppl. See http://www.pleasureridecr.com/transports and https://www.interbusonline.com . We booked on line and by phone, calling their CR local #.
Shared shuttles need to be prepaid. Unlike public buses, they deliver you right to your accommodations.
If you are travelling by bus, the site linked in Post 1, has Mepe's local phone numbers. Tickets can be reserved in advance, without paying, and purchased at least half an HR before the bus departure. Our hotel owner there did it for us, as a Plan B, in case our flight was delayed and we needed to take a taxi to San Jose and then a bus to S. Caribbean, but we made it to our shuttle (it took us 1.5 hrs and then 2 hrs to exit from SJO due to long line at the customs, etc. Rain season).
Once you make it to San Jose's Mepe terminal, the bus is about US$10 pp vs about US$54 or so for the shuttle. Note that Interbus (as well as Caribeshuttle, which we have not used) vans do not pick up right from the airport. You need to leave enough time to get out of this busy airport, plus a short taxi ride to Denny's parking lot, where some shared shuttle companies pick up. If you miss a shuttle, they wait for 10 min max., then leave, keep the $$ and you need to look for alternatives.
What time is your flight? The last public bus goes from San Jose's Mepe terminal at 6 PM, I believe, and even if you take the one before that, you will arrive to PV after dark. It gets dark before 6 pm. Make sure you have a plan how to get to your accommodations from the bus stop at the destination or have your hotel owners send a taxi for you.Edited: 23 September 2018, 15:35
We took the bus because it is cheaper than a shuttle and we like traveling with the locals. From the airport you can take a bus or taxi to San Jose. A taxi from SJO to the San Carlos bus terminal (also called Atlantico Norte) in San Jose for the Mepe bus to Puerto Viejo would be $30 or so. Or walk to the airport bus stop and take a bus to San Jose. That bus is only $1 or so. The bus dropped us off at the Tuasa bus terminal in San Jose (Calle 12 and Avenida 2). From there it was a short walk up Calle 12 (but Calle 14 works too) to the San Carlos bus terminal on Avenida 9. If you don't want to walk you could take a taxi between the Tuasa terminal and the San Carlos terminal. It shouldn't be much as it is only 700 meters or so. Don't forget you need to go to the ATM at the airport to get colones for the bus. Also, bus drivers usually don't like large bills.
There is a Mepe bus every two hours starting at 6 am. The last bus is at 6 pm. We just went to the bus station half an hour before the bus left to buy tickets but you could also ask your hotel to call and reserve tickets. It takes 5 hours or so and only costs 5,500 colones ($10). The bus is comfortable and stops in Limon for a snack and bathroom break. Your luggage goes in the compartment under the bus. Just keep a day pack with you with any valuables.