To rent a car (20 min min), drive to Jaco and return, turn car in is a bit risky. Any delays, such as a flat tire, will cause you to miss the flight. And quite a drive for just a short time in Jaco (not that great a town).

The Club is down stairs under the fast food court. It looked very nice, but I did not use it.

The Denny's is by the rental cars and Hampton about 200 yds. from airport. Just grab a ride with rental/hotel shuttle and give driver $1 or $2. A very nice restaurant and, yes, free WiFi. Also a PC internet in the hotel - usually available.

I usually go to a small motel for about $40 and take a shower and get 3+ hr. sleep. They will pick you up and return you. (about 15 min from airport). Called Riveria.
PM me if you need info.

Get back to airport at least 1 hr. before departure to do the tax free paperwork. Usually only takes 10-15 min to process and get past security - but leave some cushion.

So, lots to do. But if you are on a red eye and do not sleep well, probably best not to drive all day on strange and not the best roads.