Starling accessing the Voyager database from the stolen timeship computers may seem like simple Hollywood Hacking, but consider that the Aeon is a timeship that investigates temporal anomalies. While Braxton chose to make a more dramatic attack against Voyager on this occasion, presumably other temporal missions are more stealthy, focusing on discreetly making sure the timeline plays out as it should while making as few people in the past aware of their presence as possible. With that in mind, it's perfectly reasonable for the Aeon to be capable of hacking even technology that's centuries out of date from its perspective; it has to be backwards-compatible like that so that travellers can discreetly erase data that might affect the future.
Starling might have various other advanced technology in his office, but just because he can use it doesn't mean he can copy it. Voyager was unable to duplicate the Doctor's mobile emitter even after regular access to it for over four years, and they had access to 24th century technology; imagine how much trouble Starling might have had starting from scratch with a gap of almost a millennium to overcome, especially when he didn't seem like a genius to start with. What he introduced could have been reverse engineered from some parts of the Aeon, but at some point Starling just wouldn't be able to "risk" analysing anything else in the ship without compromising its ability to operate in the first place, "inspiring" him to go to the future and acquire more technology that he can adapt.
