Hi Susannah!
I am sorry to say that it doesn't exist any direct bus from Barcelona Airport to Valencia City or to Sagunto, although it exists a bus from Valencia City to Barcelona Airport operated by the company ALSA (the only company operating Barcelona/Valencia routes).
You have to take a bus from Aeropuerto Barcelona to Barcelona City and then a bus from Barcelona City to Valencia City with ALSA bus company. The trip betwen the two cities takes about 4 hours. There are buses all day and night and I don't think it's necesary to have a reservation.
When you arrive in "Valencia City Bus Station" you will have to take another bus LINE 115 to "Port de Sagunt" (EMT MetroBus company) wich is 30km north from Valencia. When you arrive in Port de Sagunt you have to take another bus from "Port de Sagunt" to the town of Sagunt (only 2 miles).
Here you have the links
www.alsa.es http://www.etmvalencia.es/metrobus/lin/code01/115/115.asp My advice is to take the train from Aeroport de Barcelona to "Barcelona Sants" train station. There are train every 20-30 minutes and the trip takes only 20 minutes. Then in Barcelona-Sants Station you take the EUROMED Train to Valencia it takes only 3 hours.
When You arrive in "Valencia E. del Nord" you have to take a train to the town of Sagunt. This trip takes around 30 minutes.
Please check the RENFE web. Click on Horarios to see the timetables.
Barcelona Airport-Barcelona Sants Valencia-Sagunt Barcelona-Valencia