It seems the S-Bahn line S9 to the city now starts and runs west from the new terminal, then loops round the west end of the runway and comes back through the old station at the old terminal/T5, thence onwards. I doubt there will be much transit traffic between the two terminals; more likely to be those who have initially gone to the wrong one ! It's every 10 minutes. The S-Bahn was always the most convenient way from Schonefeld into the city, stopping at all the points on the east-west line. Given the ticket system in Berlin, with no barriers and just periodic inspections on the train with fines, the most effective way to make this a free transfer is just arrange for tickets not to be inspected between the two terminals and tell transfer passengers just to get on.
I find there's also new service FEX (Flughafen Express, a polyglot mix of languages, surely it should be FSZ, Flughafen Schnellzug), with dedicated trains, which leaves the new terminal station in the opposite direction, going east, doesn't serve the old one, joins the S9 route after a distance, but then loops right round the north side of the city on the old bypass "Ring" tracks, where it makes a couple of stops, and ends up coming into the main station, the Hauptbahnhof, from the north. It apparently runs only every 30 minutes. I don't know if it charges a premium fare like the equivalent one in Vienna does, which is commonly completely empty, everyone going on the more frequent regular train, but it seems a similar style operation. Described as the fastest way to "the centre", though where the new city main station is located is about the bleakest and furthest-from-everything point in central Berlin. The S9 looks like a much more practical and generally faster way to your city destination - and for the terminal transfer.