My experience is very old and probably not that relevant (early 70s), but this is what I remember.
I attended the first three years of a state primary school (equivalent of R, Yr 1 and Yr 2). During Year 2 I sat the entrance exam for a prep school. It was a selective one - but it did have an infant dept beneath it and I think all of those children automatically passed up into the prep school. There were very few places available at 7 so the competition was probably quite high - I remember being one of only two newcomers in my class.
Obviously I don't remember the entrance exams in much detail, but I do remember it involving some reading, writing (including a story) and arithmetic, and filling in time at the end with a picture. I remember not knowing what a division sign was and going home and asking my Mum what a line was with a dot above and below it. I was not bothered that I did not know. I just did not answer those questions with the mysterious symbol in it. I remember just feeling during the exam that they should have made those questions clearer!
I certainly had written plenty of stories at school already - during Year 1 and Year 2 I had filled in spare time at school writing loads of them. (Wish my Mum had been given them). The children who had moved up from the private infant dept could do joined up writing, I could not. This cannot have mattered as I gained a place, and I just viewed it as a curious fact when I joined the school, not a problem.
I hope you are able to find out more from the school about the exam, and what sort of standard is required to pass it. I don't think all children of that age have many preconceptions about exams and whether they should be able to answer all the questions or not, or indeed whether they answered them correctly or not. I certainly did not feel under pressure to pass the exam or get a place at that school. It was just something that happened to me. I don't remember sitting in the exam thinking I want to come to this school so I have to pass this exam. I had never seen the school before.
How long to the exam? If your daughter needs to write a story in the exam it may not be as impossible as it seems. Can she invent a story of her own and tell it to you? Can she write each letter of the alphabet? Does she know all the phonic sounds in a simple scheme like Jollyphonics?
If so, you might be able to prepare in time. Get this book: How to help your child read and write, Dr Dominic Wyse, Pearson Prentice Hall
If this really matters to you, give it a shot, but don't tell your child that this is all for an entrance exam. If she loves reading, she may be motivated by making her own books - making the book, ruling the lines, writing the stories, drawing the pictures, numbering the pages, the full works. This may be sufficient incentive to want to work on specific aspects of her handwriting to speed it up and widen the number of words she can write down in a readable form.
My Mum improved my writing speed and readability by a very simple technique. She wrote out some stories by hand (her own stories, things she knew would interest me personally) on alternate lines in an exercise book. I copied them out as beautifully as I could directly beneath. This was sufficient for me, but you will find lots of other great ideas in the book I have suggested. It is not one of these silly gimmicky American ones. It is written by a lecturer in Primary and Early Years Education at the University of Cambridge.
Good luck with it all.