Why wouldn't you design it at the required size?
Depends on the printing process. You could design an A4 Document at 300dpi if your print process was 150 line screen or 150 DPI. Its about understanding how your customer is going to print the document. If its standard 4-colour process at 300dpi then you will have to design at that specification.
If you want to prepare Poster in PS. You have add some additional milimeters (3-5 mm) each side for bleeds So your document should be A4 and 2 x bleeds
Photoshop it's not a good program to prepare Posters or documents for print. Maybe (if you can) try to use InDesign or Illustrator.
In this case prepare A4 document with bleed
I'm in the process of changing my old PC, it's very slow if I start to design any bigger than A4, just thought it would be faster if the file wasn't so big.
In Photoshop, everything is measured in pixels. A large file is more pixels, a small file is fewer pixels.
Reproduction size just distributes those pixels on paper. It means nothing for file size. An A4 file may well be the same size as an A3 file - it just carries an instruction to pack the pixels closer together on paper.
This relationship is expressed as pixels per inch. That's the resolution figure, usually shortened to ppi. Read that literally! The ppi figure is self-explaining once you stop and think about it.
For a print to look good, it requires a certain number of pixels per inch. The actual figure depends on viewing distance and reproduction method. For an A3 poster it will normally be somewhere around 200-240. The required number of pixels in the file is reverse-calculated from that.