Interbus (shared shuttle)
https://www.interbusonline.com/apps/book/book.php will pick you up at the airport and drop you at your Jaco hotel. I picked a random date and the cost was $52. The only issue is that I don't know what happens if your flight is late or you get stuck in immigration. Trip should be less than 2 hours.
The is also the Caribe Shuttle for $44 but you have to board them at the airport Denny's parking lot. You will need to take a taxi or Uber to get there so most of the savings is negated and it is a lot less convenient.
You are correct, to use the public bus you need to go to San Jose centro first. You can catch a bus just outside of the terminal property to take you downtown, but you need to be sure you get on the correct bus. Personally, I would take a taxi direct to the downtown bus terminal (about $30) and the catch the bus to Jaco. Since this will be around rush hour when you get on the bus, I would expect your total transfer time to be around 4 hours or so.
If it was me I would take the easiest method and go with Interbus since this is your first trip. Keep your fingers crossed that your room is available early or worst case sit by the pool and have a Bloody Mary for breakfast.
Have a great trip