We took the bus from San Jose to Puerto Jimenez. There is a bus at 8 am and noon. Normally it takes 8-9 hours. It took 7.5 hours for us. It was $14 per person.
Yes, you can take a taxi from Puerto Jimenez to Carate. I think that would be $90 one way. Cheaper would be to take the public transport (the colectivo) which is a cattle truck with benches. It leaves at 6 am and 1:30 pm from Puerto Jimenez and takes 2-2.5 hours and costs $9 per person one way.
If you will spend some time around Puerto Jimenez as well I highly recommend taking a trip to Dos Brazos de Rio Tigre. Jungle Hostel Bolita is amazing. You have to hike 30 minutes through the jungle and wade through a river to get there. Lodging is basic and cheap but the trails are spectacular. There is other lodging in the village as well and you can still hike the trails at Bolita for $8 if you are not staying there:
http://www.dosbrazosderiotigre.com/accommodation.html
You can get to Dos Brazos by bus on weekdays (11 am and 4 pm, 40 minutes, $4). Or by taxi ($20).