There is a line of "reasoning" I sometimes hear from people in the SCA: "well they had indigo dye, and they had french seams, and they had pants, so blue jeans are totally period!" Well, yes and no.
So too with modern music forms. Yes you can trace some elements back -- they had drums in the middle ages, and recited poetry with instrumental accompaniment, and certain musical forms like isorhythmic motets were all about rhythmic presentation, and you can squint and tilt your head sideways and say "sounds like rap". But a musical style develops in a context, so you would need to find some reason in your world for these particular events to come together in this particular way to produce a given result. What problem is solved by blue jeans? What development is facilitated by rap? Can you find that need within your world?
One path you could take: not everybody is skilled with musical instruments, and not all of them can play drums either (trust me...), but rhythmic accompaniment is a lower bar to entry than the lyre or the lute. So you could imagine a setting in which a drum-accompanied alternative to the lyre-accompanied poems arises, because that's what people in that small village know how to do. From there, you might find the texts evolving, perhaps to emphasize simple rhythms better. Throw all that into the pot for a while and you can see how something kind of like rap could emerge. Or you could start from the aforementioned isorhythmic motets and start simplifying the melodies and reduce them from 2-3 parts down to one to accommodate the skill of the singers you have, and then go from there.
But it will only be believable if it arises from within your world, not if it is retrojected into it.
(By the way, medieval and renaissance music is one of my hobbies.)