Yes, these two terminals are direcltly connected, and most probably no need to recheck luggage.

However, you will be going thru passport control as you would be exiting the Schengen area (the terminal where your FRA flight landed) and going into the non-Schengen area (where your next flight leaves).

This is rather new (BUD has been in the Schengen area since late March) so I don't know how is this working and what is the min. connection time between the two areas, but most likely will not be a problem.