Sometimes these things go so well you wonder what al the worry was about, sometimes it ends in disaster. You just don't know until you try. I also strongly believe the more room you have the better your chances. That room is inside the coop and outside when they have access to outside. There is nothing wrong with trying it this way, but I'd want the coop dark enough at night so the others can't beat her up and I'd be down there at the crack of dawn to see how it is going until I'm comfortable she will be OK.
Don't be surprised if she is not cuddling up to the others and joining in all their fun. When I integrate young ones they almost always avoid the older ones. If they invade their personal space they are likely to get pecked so the quickly learn to stay away. When I go down to the coop in the morning to let then out I typically find my juveniles on the roosts while the adults are on the coop floor. The juveniles are avoiding the adults. If she wants to sleep on the floor or somewhere else instead of on the main roosts with the others let her. I don't care where my juveniles sleep as long as it is not in a nest and is somewhere predator safe. It really helps if they do have room to avoid.
A potential problem with a single bird is that they are social animals, they want to be with other chickens. Sometimes the older ones will let them, it is possible. But sometimes the younger ones have to hang on the outskirts of the flock until they mature enough to actually join in. Integrating one bird can be harder, but it can be done. If you see this, be patient. Don't try to force them to be any closer together than they want to be. This can last until the younger starts to lay but sometimes it works out younger. There are no guarantees with any of this. Sometimes adolescent birds can be worse than adults, sometimes they can be welcoming. You never know until you try.
If this does not work as you plan, I'd set up a separate place you can keep her near the others for a few days so they can see each other. Then once she knows to return to that place to sleep try letting her roam with the others outside during the day when you can observe. After they've proven they can coexist during the day without issues you can try moving her into the main coop if you wish.
Again I'll mention patience. My goal during integration is that no one gets hurt. That's it, no one is hurt. All that one big happy flock doing things together will come with time, but often it takes time.
Let us know what you try and how it goes. If you have problems information on the size and layout of your coop and run and photos showing how they go together can be very helpful in giving specific advice or suggestions. It's a lot easier on us if we know what you have to work with.
I definitly understand! I am going to keep my plan, and I will be out at the crack of dawn like you said. I also have a chicken with a lame leg that could keep the new one company.
Here is my plan:
Pick chicken up in early morning of saturday
Let my chickens start free ranging with supervision. I will line feed up by the edge of the fence and put the new chick in the coop.
When they finish eating, I leave them to interact with the chick through the run for most of the day.
When afternoon comes around, I'll put my lame-legged chicken in with the baby w/ supervision and see how they act. I leave them alone for 2 hours.
Then after those two hours I block the chicken off in a corner of the run with chicken wire and supervise.
At nightfall I put the chicken in my room.
At 9-10pm I sneak the chick onto the lower roost, where no chickens sleep. (If she jumps off I'll leave her)
On sunday morning, I wake up early and let them out with supervision (2 feeders and 2 waterers will be available)
If the chick is bullied too much, I will remove her. After everyone seems to have eaten, I will let them free-range. The rest of the day I supervise.
That is the plan

And I will submit photos of the coop and run from my phone!
(keep in mind I will add another feeder and waterer they just will not be pictured in this photo)
(In spring I am also planning on buying a rabbit hutch for the chickens to spread out but for now the chick is small and it is cold so I am thinking it will be fine)
(And I free range them everyday when I get home)