How did Leia manage this feat?

Leia used the force, brought out by her survival instinct.

Survival in space is something that seems within the grasp of Jedi to do, as is evidenced by Plo Koon in The Clone Wars

I can withstand the pressure for some time

While it's unclear whether or not this is a skill specific to the Jedi, it seems to be with their grasp. Although Leia is not a Jedi, she is evidently force-capable and may have been able to use the force to keep herself alive.

Rian Johnson confirms that this is something to do with the force, related to a survival instinct, like Luke in ESB

I liked the idea that it’s not Luke concentrating, reaching for the lightsaber; it’s an instinctual survival thing, like when you hear stories of a parent whose toddler is caught under a car and they get superhuman strength, or a drowning person clawing their way to the surface. It’s basically just her not being done with the fight yet.

I wanted it to happen [for Carrie] and I knew it was going to be a stretch. It’s a big moment, and I’m sure it will land different ways for different people, but for me it felt like a really emotionally satisfying thing to see.
_{Inverse - ‘Last Jedi’ Director Explains All the New Force Powers (and the Leia Scene)}

This is re-confirmed in an interview with IGN:

“This is a reflex action on her part,” Last Jedi writer-director Rian Johnson explained. “It’s the equivalent in my head of when you hear about parents, toddlers are caught under cars, and they suddenly get Hulk strength and can lift it up. Or a drowning person climbing their way to the surface." ...

“It’s instinctual, her use of it. It’s the opposite of when Luke Force-pulls the saber in Hoth. It isn’t like, ‘I’m going to try and do this.’ For her it’s just an instinctual thing of, ‘I’m not done yet. I’m not giving up. I’m pulling myself back in.’”
_{Star Wars: The Last Jedi Director explains Leia's relationship to the Force.}

Creatures such as the Purrgil were able to survive the vacuum of space even though they weren't Force-Sensitive, how they could isn't clear, but they were also able to jump into hyperspace.

Out-of-universe

After Carrie Fisher's untimely death, Kathleen Kennedy and Rian Johnson had discussed changing the scene to make it so that Leia died "drowned in moonlight" as she had so famously wished for, however they had decided not to change the scenes and to leave the survival intact.

“We were a little ways into postproduction when she passed away, and so we had it mostly put together,” Johnson said. “We didn’t really end up changing it. And that includes adding lines back in that we had cut out or anything like that.”
_{Vanity Fair - Star Wars: The Last Jedi—What Happened to Leia?}

The decision to not make the death so in-line with Fisher's iconic quote may have changed the beauty of the scene out-of-universe, but Fisher still had a role to play In-Universe. It also makes it clear that as beautiful as the parallel is, it’s not certain that this was the intention of the scene, as it had been filmed before her death.

Short answer: We don't know

Long answer: That scene was more of tribute to Carrie Fisher's joke about her wishful death. From vanityfair:

The image can’t help but recall the late Fisher’s famous wish for her obituary, from her one-woman show Wishful Drinking. Recalling how Star Wars director George Lucas requested she go bra-less in the original trilogy, Fisher once joked:

George comes up to me the first day of filming and he takes one look at the dress and says, “You can't wear a bra under that dress.” So, I say, “Okay, I’ll bite. Why?” And he says, “Because . . . there’s no underwear in space.” What happens is you go to space and you become weightless. So far so good, right? But then your body expands??? But your bra doesn’t—so you get strangled by your own bra. Now I think that this would make for a fantastic obit—so I tell my younger friends that no matter how I go, I want it reported that I drowned in moonlight, strangled by my own bra.

But they later decided to keep her alive. From same source:

But director Rian Johnson and Lucasfilm president Kathleen Kennedy eventually decided to leave Fisher’s role in The Last Jedi untouched. “We were a little ways into postproduction when she passed away, and so we had it mostly put together,” Johnson said. “We didn’t really end up changing it. And that includes adding lines back in that we had cut out or anything like that.”

As far as I was aware, although Princess Leia was Force sensitive, she hadn't undergone any training to put these powers to use and/or couldn't use any Force powers.

There are about thirty years of time which we were not shown, in which her brother created a Jedi academy, she had a child, her child went to the academy, destroyed the academy and Luke went into exile. That's an uncountable amount of moments and interactions between characters that none of us are aware of, detail-wise.

Seeing as how in episode #6 he and she both came to know that they were siblings with a powerful inherited lineage in the force, it seems as likely as not that Luke would have shared at least some of that knowledge with his sister in the decades between their father's death and Luke's voluntary exile. A time when his mission in life was to teach the ways of the Force to others.