It depends on which airline you’re flying with. Costa Rica has two airlines with a domestic flight service: Nature Air and Sansa.
Nature Air policies:
-40 lbs per person for check-in baggage free of charge
-One carry-on handbag up to 10 lbs free of charge
-One piece of luggage with a maximum weight of 25 lbs and a maximum size of 45 linear inches (length + height + width)
-One carry-on up to 10 lbs
-Each additional pound is charged at $1 per pound
-Total baggage weighing over 50 lbs will be charged an additional fee: $7 for 1-5 extra pounds, $20 for 6-15 extra pounds, $40 for 16-30 extra pounds, and $60 for 31-60 pounds.
Neither Nature Air nor Sansa have overhead compartments, so your carry-on must be really small.