The first time Alex sees Kara's spaceship, she doesn't realize what it is. It's not until Hank leads her over to it, says pointedly, "a Kryptonian craft that landed here 10 years ago," that she recognizes it, sees the similarities to the drawings Kara had done when she was younger.
Those first few days at the DEO, Alex takes any opportunity she can to get close to the pod, to touch it or clamber onto it when no one is looking. At first it's just a way to feel close to Kara. They're in different cities, and Alex is all alone with another secret to keep. Soon though, the scientist and engineer in her is begging for a closer look.
Of course, the DEO scientists have studied the spacecraft. They've turned it inside out, removed pieces and put them back in (wrongly, Alex discovers many times). But Alex grew up with a Kryptonian, knows the ways she thinks, how she moves. Knows what is instinctive to Kara, what would seem backwards to humans. She figures out how to power it back on the first week (it freaks everyone at the base out, especially Hank. Alex assures them that it's just enough power to turn the lights on, that the pod will never fly again (that's a lie, she totally thinks she can make it fly)).
Alex finds Alura's AI program by accident during her third month at the DEO. She's taken to eating her meals on or in the spaceship. It's a common enough sight that people have stopping talking about it (at least where she can hear them). She's staring at a small screen on the dash of the pod, snacking on a protein bar as she absently swipes through multiple screens, when she notices a small House of El symbol blinking at her. She touches it, and a small hologram of Alura appears in front of her. Alex swears and jerks back, banging her head on the roof of the pod and smashing her protein bar into the screen as the hologram begins to speak.
It takes a few days, between Alex's slightly rusty Kryptonian, and Alura's refusal to answer certain questions, but Alex eventually figures out that the pod's systems contain a flight simulator. From there it's only a couple of weeks before Alex has the spaceship fully functional once again. The AI confirms that there are populated star systems and planets that are within the ship's reach, that Kara, as daughter of the House of El, will be able to command the pod to take her there.
Alex is relieved, knows the other agents can tell a difference in her, catches even Hank looking at her quizzically, but none of them know. They don't know the plans Alex has been making for the last 10 years, how those plans have changed and matured as Alex has (the very first plan involved glasses and a wig, and hiding in the woods; the more recent incarnations consist of fake IDs and cash. Lots and lots of cash). They don't know the burden that has been placed on Alex, how those plans (never good enough, never safe enough) have weighed Alex down, have haunted her sleep. But now, finally, she's found a way for Kara to run, a place for her to hide where no one can follow her. Because maybe, someday, someone (the wrong kind of someone) will find out what Kara is. And if they come for Kara, Alex will protect her, will send her away, will break Kara's heart and her own if it means her sister will be safe. For the first time in a long time, Alex feels peace.
After almost half a year in the DEO, Alex is told about Fort Rozz, and learns that there are other Kryptonians on Earth. That night she wakes in a cold sweat; dreams, not of Kara being forced from her, but of Kara leaving willingly, following her fellow Kryptonians to a new home among the stars.
Alex is tired, the next day, then the next month, as variations on the same dream visit her every night. Other agents start avoiding her and Hank asks if she's okay. Alex shrugs him off, spends the rest of the day staring at the spaceship before finally venturing in. The AI is reluctant at first (petulant, Alec calls it the first few weeks), but eventually recognizes Alex as an ally of the House of El and gives her full access to the ship's mainframe. Alex returns home that night and falls into a restful sleep for the first time, because now, if something ever happens, if Kara returns to the stars…well, now Alex can follow her.