I'm looking to shift from a modern world to a fantasy world. The modern world has some magical elements to it, such as remnants of the religion that reigns in the fantasy world, but is mostly magic-free, so I kind of shy away from using something like portals. What are some other ways that my character can get from World A to World B, that aren't overtly magical?
If you need to get from one world to another without magic, a spaceship is the way to do it. FTL or hyper-sleep, take your pick.
It's a portal made out of science!
In all these cases the fantasy world is not real.
It can be done fantastically as in The Wizard Of Oz, as a mystery as in The Matrix, as delusional escapism in Pan's Labyrinth and Heavenly Creatures, or tragically as in Jacob's Ladder where...
...the protagonist is slowly dying in reality.
Just Do It
Don't explain why the world is fantastic, it just is. This doesn't have to be overtly fantastic. For example, a musical is a parallel world where people spontaneously break out in song and choreographed dance with strangers. Or a comedy where slapstick and mind-bogglingly dumb decisions are simply accepted as normal. For example, Who Framed Roger Rabbit asks the audience to accept a gritty 1930s noir setting that also has real cartoon characters, and it works. If the characters accept it as reality without question, and it's well fleshed out, the reader will as well.
You could have her not go. Have the magic world revealed to her as existing on, beside, in between the one she knows. My favorite example of this is from a fairy tale where a midwife is summoned by a dark man. After a trip through the night with his hands over her eyes they come to a castle.
/At last we came to a bedroom, with a beautiful lady in bed, with a fine bouncing boy beside her. The lady clapped her hands, and in came the Dark Man and kissed her and the baby, and praised me, and gave me a bottle of green ointment to rub the child all over.
'Well, the child I rubbed, sure enough; but my right eye began to smart, and I put up my finger and gave it a rub, and then stared, for never in all my life was I so frightened. The beautiful room was a big, rough cave, with water oozing over the edges of the stones and through the clay; and the lady, and the lord, and the child weazened, poverty-bitten creatures—nothing but skin and bone—and the rich dresses were old rags./
From The Lilac Fairy Book, Andrew Lang. http://www.sacred-texts.com/neu/lfb/li/lifb07.htm
The fairies don't like that she can see them as they are, and many versions of this tale end with her having the anointed eye plucked from her head. The idea of the magic world and the mundane overlapping is a common one, from fairy tales like this to Mary Poppins to Harry Potter. It lets you sidestep some worldbuilding by simply augmenting the real world.
On the physics, it is impossible. The laws of our Universe are simply totally incompatible with any fantasy-like world. Practically any "speciality", magic-like thing what in the RPG worlds exist, is totally impossible on the current laws of the physics.
Our world is a perfectly controlled one, there is no place for doing anything by pure imagination, only by clearly applied rules.
But we can do some similar:
New realities are imminent: how VR reframes big questions in philosophy
The virtual reality (VR) industry is currently in its infancy, but in just a few decades it’s possible that virtual environments will be nearly indistinguishable from reality. Along with transforming everyday life, a VR revolution could fundamentally change how we understand and define what is real. In this installment of Aeon In Sight, the renowned Australian philosopher and cognitive scientist David Chalmers considers how VR is reframing and shedding new light on some of philosophy’s most enduring questions about cognition, epistemology and the nature of reality.
...and, a possible horror-line to the story: consider a Humanity, who lives in a continuous dream, and machines serve all their bodily needs. Only a small group of "service workers" are watching and controlling these automatas. Yes, it is a little bit similar to the Matrix, although it is a much older idea and the "internal reality" is a fantasy-like one.
Considering the intensity of the MMORPG-addiction in our children, such a world may be not even a too far future.
In this world, in the reality, there is actually nothing. Old cities, full with people in large, cubical buildings, in a continous dream. And out of them, nothing.
These "service people" will obviously feel an increasing urge to join the others in the virtual reality. The external world is boring for them.
And, as years... decades... maybe centuries pass, once the time is coming as also they lie beside the others, and forgot the external world.
I'll try give some more interesting ones:
Kind of like Dr. Strange's astral projections. If you focus on your breath, get your frame of mind just right, you can open your eyes and suddenly you're in the other realm.
Did you ever play with that book when you were a kid, that shows a random pattern, and you have to unfocus your eyes in a certain way to see a 3D effect? You could do something like that, which shows you the other realm.
Instead of having them in different times, they could exist simultaneously but separated by a giant game of thrones style wall (but one that nobody has ever thought to climb/cross). Your protagonist somehow finds a hidden tunnel. No magic here, just a plain ol' door.
To transport someone from one world (which I'm assuming means "planet" since you asked for non-magical means) to another, there are basically two options: spaceships and portals/teleportation.
So I guess it depends on the technology level of your modern world. If it is high enough that spaceships are common enough for them to be a viable means of transport (as in <hopefully> our world in 10-20 years), then that is likely the best method. If spaceships don't exist, are only uncrewed, are used solely by government entities, etc. (similar to our present-day world), you'll probably have to go with portals or teleportation of some sort.
There are two types of portals: scientific ones (wormholes) and magical ones. Unfortunately, wormholes require an even higher technology level than spaceships, and are therefore probably infeasible. Magical portals have practically zero constraints, because they're magic. The same goes for teleportation: scientifically sound teleportation is (as far as we know) impossible, but you can do almost whatever you want with magical teleportation.
There are other options that could work non-magically, depending on your setting. For example, if your worlds are orbiting each other very closely (i.e. closer than Earth's geostationary orbit) and are tidally locked to each other, a space elevator could suffice, and would likely be possible with present-day technology.
That's about it, unfortunately. There are plenty of overtly magical options, but not much that could work without magic.
Use A Boat
Your planet has two big continents, separated by a big ocean. If you need a faster transition, there could be a land bridge. They could both have diffrent climates that allow for different species. The tech/fantasy continent could be the only one capable of making boats if you want. This also allows for no one knowing of each other's existence or just one side knowing. You could also could have someone attempt to sail around the world only to accidentally find the other continent. There are a lot of diffrent things you could do in this situation.
I think it would be especially cool if the device were mostly technological (with a little magic) on the modern world’s side, and essentally magical with some little technology (like precision instruments) on the fantasy side.
It each end of the transit mechanism works by the rules of its respective universe.
The whole thing is like impedance matching, in a sense. The two independent efforts to reach another universe were mutually successful, and connected to each other.
So maybe scientists on the modern end were trying to teleport to Titan, boosting their experimants with magic spells. In the fantasy realm the mages had a similar goal that makes sense to them, and they added some precision machinery into tneir magic device.
Person dies and is reincarnated in another world. You could put in something like karma, where the body they go into is based on their past deeds.
Similar to in Kimi no Na wa, the person goes to sleep and swaps body with someone else in another world after they wake up. They don't even have to be in the same timeline. Someone could swap bodies with someone who existed 1000 years ago.