You don't have to do or pay anything, the local connection part of Sparpreis/Saver ticket is flexible.
from the DB website english pages:
*If you have a (super) saver fare ticket, you can only travel on the service indicated on the ticket. Offers are available for ICE/IC/EC connections within Germany.
You can use local transport (e.g. RE, RB, IRE and S-Bahn or non-DB trains) to get to/from a long-distance station if you include them in your booking. You are not restricted to travelling on a specific train when using local rail services. However, your journey must be completed by 10:00 am the following day.*
The booking system, like most others, is biased to giving the overall fastest intinery which is why it offers short connections, clearly if you used a later S-Bahn and missed the ICE they would hold it against you but going earlier to avoid a rush is fine. In fact you can also use the slightly better RE trains if one turns up first. I think all trains to the Hauptbahnhof go from track 1 at Regionalbhf - they don't stop long so don't wait for instructions. Note that virutally all stations in the Frankfurt area have names begining Frankfurt, you need Hauptbahnhof (abbreviation Hbf) . Most S-Bahn trains continue beyond Hbf and will show either Hanau or Offenbach on the front
Be aware that there are two stations at the airport, for S-Bahn you need the Regionalbahnhof which is under the main drop off/pick up road outside Terminal 1, entrances are marked with circular green signs with a white S. Do not go on the hike down the corridor to the Fernbahnhof on the other side of the autobahn!