I know we all are basing our understanding of Anarchy off SR5, which in most cases is understandable since Anarchy is so loose and undefined.
Matrix, Magic, and Combat actions generally take one "attack" each. So what's an "action"? Who knows; whatever we think is reasonable and semi-realistic. Movement only matters in the context of changing between Close/Medium/Far. Everything else is swept under the rug of Narration, including details like going prone or standing, picking up or dropping things, taking cover, etc.
I think something in Anarchy says to think of a Narration as lasting a few seconds, but I don't want that to limit our roleplaying. Maybe don't give a full Braveheart speech during a Narration - you'll have to summarize it - but I'm not going to make people count words like I do in SR5. Similarly, if it's cooler to narrate a longer back-and-forth of fisticuffs or gunplay, by all means do so even if it wouldn't logically fit into three seconds.
When it doubt, feel free to ask. But, know that one of my primary reasons for doing this scenario in Anarchy was to gloss over a lot of the nitty gritty details. Who wants to play out an attack a full platoon of soldiers via PbP with SR5 rules? It would take a year! When in doubt, keep things moving forward.
So, what can Shiriki do. He can issue a basic order to the Dragon as an "attack" (which needs to be resisted by the device/host), but the Dragon still needs time to execute those orders. It can't go from being on-the-ground-with-engine-off to in-the-air-with-motors-spinning within one Narration. Let's break that into two, where this Narration it fires up and next Narration is takes off, at which point Shiriki will pilot it clumsily.
That does leave two actions for Shiriki to hack something else, or to jump into his drones and open fire. They can move from Far to Medium before shooting.