The RMV is the transit authority that covers ticket sales for local/regional transport on your route between Frankfurt Airport and Mainz (or Bacharach.) The regional transit authorities in Germany operate much the same everywhere... in addition to bus, tram, subway tickets they also sell tickets for REGIONAL TRAINS (not the ICE, which DB sells) for journeys within their jurisdiction - and one ticket is valid for all these local forms of transport, in whatever combination is necessary for your journey. So if one's destination hotel lies far from the station, one RMV ticket with that hotel's address as your destination will typically cover both train and bus (if a bus route is available.)
Map of RMV train lines
RMV rules of travel... Any of the regional trains (RB, RE, S-Bahn) can be used with the RMV ticket. From FRA, RMV point-to-point train tickets to both Mainz and Bacharach are bought like bus tickets - on the spot, on the day of travel; tickets must be used on the specified travel route without detours or long stopovers and within a very short time frame after purchase. Deviations can result in an invalid ticket and fines. So it's in your interest to purchase your ticket just prior to boarding.
DB (Bahn.com) also sells regional train tickets - but only for journeys that involve travel through multiple transit authorities. If you were traveling not to Bacharach but 5 minutes further to Oberwesel, you'd be traveling outside the RMV's zone of authority, and a point-to-point DB ticket would be required instead. DB tickets are much more flexible. You can use the exact same trains, but you can buy the ticket on any day you like for the specific travel date you have in mind, you can use the ticket at any hour, and you can make whatever stopovers you like. These tickets can be purchased in advance, but there is no price advantage, and since seat reservations are not available anyway, there is usually no reason to pre-purchase these tickets, or ANY regional train tickets or day passes. They are always available on the spot.
"I think the regional day ticket is what I need."
The RMV regional group day ticket (€37.50) costs more than point-to-point but provides time flexibility. (Like the other DB and RMV regional train options, there's no advantage for purchasing the RMV group day ticket in advance, as a rule.) That said... The upcoming €49 ticket: It's not clear what your plans after Day 1 will be. But if you plan to travel by regional train on subsequent days as well, this ticket, valid Germany-wide, could be purchased on Day 1 and would include the journey to Bacharach (no RMV ticket required, no R-P ticket required.)