i think it has something to do with the placement of 못 that determines whether it means you are "not physically able" to ride a horse for the duration of an hour, or it needs a different word to say that you can't ride a horse until an hour is over.
TBH i think it's the first one. that "I cannot ride a horse for the duration of one hour"
Logically speaking it could be both, but practically it is the latter. If you are saying you cannot ride a horse for an hour straight, it implies you can ride a horse but only for like thirty minutes or so. Thus in this case you want to use the particle "는".
저는 말을 한 시간 동안은 못 타요.
저는 말을 한 시간은 못 타요.
저는 말을 한 시간 동안 타지는 못해요.
In English "during" indicates within a period when some condition is true. "For" indicates a length of time. E.g. 1) "I study during the day" (i.e. not at night). E.g. 2) "I studied for a day" (an entire day). You can ride a horse for an hour but "during an hour" (which hour?) doesn't make sense.