You might get a set of magnetic letters and a board to go with them, and spend a few minutes every day (maximum five) moving letters around on the board together. I would not start with a sentence, though. However, your rhyming words (fat, cat, rat, etc.) would be fun to play with. I did this with my children and as long as the child is enjoying it, and you aren't pushing, why not? Once you've got the A and the T set up, you can slide different consonants in front of those, and read the result together.
You might look into some Montessori activities, such as sandpaper letters. (I don't know a lot about that.)
(I'm assuming you read several children's books to her every day.)
You said she is interested in what she sees you writing in your notebooks. This is a great place to start! Give her a notebook like Mama's, and let her choose what she wants to use for "writing" in it -- pen, pencil, crayon. Sometimes let her write isolated letters or sketches or scribbles, play-acting "writing like Mama"; but sometimes also take dictation from her, in HER notebook. When you write what she dictates, write in large, clear print, and read it to her as often as she likes.
You can also make other media available to her, such as an easel with paper on one side and a chalkboard on the other. For variety, let her sometimes draw, write or scribble with different kinds of paint, including finger paint.
As she begins to write, allow her to use "invented spelling." In the early stages, don't correct her spelling. Here is an article that explains invented spelling (also known as inventive spelling) and the various stages of learning to write: http://www.readingrockets.org/article/invented-spelling-and-spelling-development
Four is quite young. Every child is unique. Some children are ready to read and write at four, but most aren't. Please don't be disappointed if she's not ready to start writing yet. Even if she isn't, you can be pleased that she is interested. If you make sure that her experiences related to reading and writing are pleasurable and stress-free, that will be a strong foundation for future intellectual accomplishments!
Another thing you can do with a pre-reader is math, including geometry -- for example, a four-year-old can already be having fun figuring out which shapes are symmetrical. You can even explore different kinds of symmetry. Also, you can start skip-counting now -- counting by two's, three's, five's and ten's now, as part of your games and activities. And you can start playing simple card and board games now.
A beginning reader often makes best progress with repeated reading of a short homemade book that is about her and her family and pet if she has one. This is something a parent can offer that a school generally doesn't.