Blinking question mark on folder on boot I am attempting to boot up my 2009 MacBook Pro, but when it starts it immediately shows a white screen with a 'question mark folder' — it keeps blinking until the computer shuts down (unless I keep it in the charger, then it just goes on).

I have tried to do the start up whilst holding down the option/alt button, and only the arrow comes up. I don't have the startup system disk (or any other) but the system has been updated to OS X Mountain Lion from the Mac App Store.

Before this, there had been other problems: when I deleted things they didn't go in the Trash but instead it looked like it just deleted it completely. After some time it wouldn't save or download, saying there wasn't enough space (even though I deleted items).
If only the arrow comes up when you hold down option it most likely means there's some kind of hardware issue.  You may be able to repair it using the Disk Utility in Internet Recovery (Hold down ⌘ command+⌥ option/⌥ option+R while booting) , but I'd advise in to looking to getting this thing serviced or repaired.  Sounds like you've got a bad HD.  

If all else fails, run fsck and see what happens.  Information on how to do that is located also in the provided article.