Sintra is a provincial town and has no airport. That is why to get to Barcelona, you need to reach a larger city close to Sintra first, from there you'll be able to fly.
It is impossible to buy direct flights Sintra — Barcelona, because the former does not have an airport. However, pretty close the following airports are to be found:
Find the best accommodation in Barcelona on hotels.anywayanyday.com: search for hotels by price, popularity, trip ranking, accommodation type and rendered services.
