I've just had the same problem. First iTunes would not allow me to backup my iPhone due to not enough room on my C drive.(Windows 7)
I fixed that problem by following the exact instructions as follows:-
But then i had the same issue of not having enough space on my C drive to download the updated IOS.
Basically you need to follow the same process, except the route/path to where iTunes downloads the IOS update is of course different to the backup folder location.
Instead of pointing to a folder called Backup in
(as described in the How to geek article), you need to do the same process to the location of the Phone Update folder which in my case was;-
and look for the folder called "Phone Software Updates" (and follow the same process as per the how to geek article).
As long as you are careful and consistent this worked well for me. Don't be put off by the fact that the command that you type to make the link to the new location (in my case was my E drive doesn't seem to fully describe the path by omitting in my case the "\roaming\" part of the route/path.
What you are doing is replacing the folder on your C drive where iTunes thinks these files are stored, with an identically named folder which instead points to an off-board drive by using a symbolic link Directory Junction.
I apologise if i haven't been able to explain things particularly well, but computers are not my expertise, and this clever little fix was (and still is) way beyond my knowledge without some assistance. BUT it did work, and i kinda get the basic principle and applied it consistently.
So i was successfully able to do a backup and then upgrade my IOS