That sad task is one of those "practical" things not often written about, so there's not a lot we can say definitively about how it was done in the very broad span of time and place you ask about. However, we can make what I think is a pretty good guess.

Ritual sacrifice is perhaps our best attested source for the slaughter of livestock. The Romans were quite interested in scenes of sacrifice, as it was a major component of their religious practices, but they preferred to focus on scenes around the act itself; there are lots more bulls being led to slaughter than being slaughtered. The preparations may have been to them the more important part (and certainly they're the part where the important people who commissioned the artwork have important things to do.) But that's an aside.

While on occasion an axe or large hammer (perhaps for occasions calling for no blood) are attested, by far the most common method was to cut the animal's throat with a knife. Both hammer and knife are still used today for animal slaughter.

A knife being a likely tool at hand for a rider, and sledgehammers being fairly rare on the road, I expect that's what they'd use. While a modern person may find it hard to believe, most historical people would have thought it quick, easy, and fairly humane to cut the animal's throat.