It's an hour drive to Poas from Alajuela; if you don't have a car, you will have to adjust to bus schedules, so don't see how you can visit Poas in the a.m. and still stay on schedule to catch a bus to Puerto Viejo. There might be a tour that doesn't spend extra time at another stop.
(p.m. won't cut it; the park closes pretty early, and the clouds move in late in the morning and you may not be able to see the caldera.)