It's important to note that the skulljack has 2 uses in the game. It's initial use in the game is purely for advancing the story, using the skill called "Skull Jack".
It's second use is to hack enemy Advent soldiers for non-story progression purposes. You can't do this until you research the "Skull Mine" ability in the Proving Grounds. Once you've researched that, the Skull Jack will increase the hacking ability of any soldier it's equipped on by 20 points.
Skull Mining an Advent soldier might help speed research times. It depends on what the hacking rewards are each time you hack a new soldier, and the rewards are randomly generated. You might get lucky and find a reward for reducing the time on your current research project, but I haven't confirmed that that's an available reward for hacking soldiers (it might only be for comm relays or chests).
The easiest way to speed up research times is to purchase research bonuses from the Black Markest, which can half the remaining research time for various projects.