So my PC in my desk is getting quite hot due to no air flow, it would be a lot better on a shelf above my PC but question is will it be ok on a shelf bare in mind this wall is plasterboard. My case is only 23cm wide so most shelfs will fit it 

Where is it located now? You say "In your desk". Rather than mounting it on the wall and worrying if the wall can take it why not just put it on a side table or something similar. Getting it off the floor and giving it room to breathe is all you need.
Depends on the PC, case width is less important than the amount of KG's you want the shelf to hold. I'd be more looking at supporting it on the table than the wall.
Ive put bare pc components on wooden slats before, as its 6 foot off the ground Im not sure it needs a case so much.
What spec and temp is it
I used to have Dusty Springfield records on mine.
But now that I've sold them I just don't know what to do with my shelf.