Propulsion tech sure is advanced by the early 2030s, judging by the line in VOY "One Small Step" about the stranded crew of Ares IV only having to wait two weeks for a rescue ship to get from Earth to Mars.
Unless the "rescue ship" was simply the next scheduled ship that was going to arrive in two weeks anyway, repurpose as a rescue mission.
Maybe the Botany Bay just had an oddly shaped centrifuge interior and artificial gravity was created the old fashioned 2001 or B5 style? SCOTT: Well, it looks like that ship was expecting us. We read heat coming on, complete oxygen atmosphere. Considering how the TOS producers emphasized the vital role of artificial gravity in the Writer's Guide and especially its limited availability while on battery power only ("In case of total failure of all engine power sources, the vessel's gravitational and life support systems can be switched to battery power, with a full-load capacity of about one week."), add to this that the Botany Bay apparently only had nuclear power (the equivalent of a 23rd starship battery?), it looks like it had to rely on centrifugal AG. Bob