We did the post-tour extension to Tallin and Helsinki and did not need a transfer to the St. Petersburg airport. That said, I suggest seeking your Program Director's advice. If they have others departing on a similar schedule and have room, he/she may include you on the bus to the airport. If not, he/she or the reception desk may make arrangements for you with a private car or taxi.
The dock GCT uses in St. Petersburg is some distance from downtown St. Petersburg and not the one used by large ships cruising the Baltic. I seem to recall some taxi activity at the dock and there is a retail shop that might be helpful in securing a taxi.
If you wish to make arrangements from home, do a Google search for "St. Petersburg Russia Airport Transfer". I've no experience with it, but the following looks credible:
If you choose to make a reservation using the Internet, I suggest confirming it by eMail or telephone when you arrive in St. Petersburg.