All you need is take the electric train from Moskovskij railway station in S-Petersburg up to Volchovstroj station. It will take near to 2 hours for the trip, if the train goes all the time, of course. But if it stops for the long time on every station for unclear reason, trip may be last three hours or more.
After arrival to Volchovstroj city you need to traverse the vast railway station square and try to take the bus #23. Bus #23 is synchronized with arrival of electric train usually, but as electric train is not synchronized with bus #23, so there are not any guarantees.
The way to Staraya Ladoga is about 10 km length, ticket costs 8 rouble. Good luck!