Between the airport and St. Petersburg run buses number 39 and number 39E, as well as minibus number K-39; the fare is 40 rubles.
To get from the airport to the historical center, take the bus / minibus and get off at the stop "Moskovskaya metro station" and change to the metro.
To get from the historical center to the airport, take the metro and get off at Moskovskaya station; further - by bus / minibus to the stop “Pulkovo, new terminal”.