Cerro Lodge is about an hour or so from the airport and there's an excellent range of species nearby (though short taxi rides are helpful):
Not sure whether they do airport transport. It's not far from a main road so there should be buses but you'd probably need a cab for the last bit (it's down a fairly obscure road).
For a contrast, I liked Paraiso Quetzal in the mountains south of San Jose. Would suspect you could get near to it by bus because it's just off the main road. Very nice forests and grounds on site with lots of higher altitude specialities. The climate is very pleasant in the day time, but cool at night.