Make sure that fast startup in the Windows control panel is disabled. If it is, shut down Windows. After booting to Ubuntu, try to mount the partition. If you still can't mount it, try to do it using the command line so you can see what's happening. Find out the partition identifier and use this:
mount /dev/sdXY /media/win
(replace X and Y with the identifier)
If you see something like this:
Please shut down Windows fully (no fast restart or hibernation).
then it means you need to shut down Windows without fast startup. If you still can't mount the partition even after shutting down fully, without fast startup, then you'll need to use NTFS since it has better support than FAT32 and there are some tools that will help you.
If you have NTFS:
sudo ntfsfix /dev/sdXY
Again, replace X and Y with the identifier.
NOTE: Backup your data in the NTFS partition before doing that. Also, you should create a restore point if Windows fails to boot. Use this method only if you have disabled fast startup and shut down Windows fully.
This method worked for me without data loss, but you should always backup your data.
How to convert FAT32 to NTFS:
Boot Windows and open CMD.
Use this command to convert a FAT32 filesystem to NTFS:
CONVERT X: /fs:ntfs
Replace X with the drive letter you want to convert. It should work for C drive (if you're on XP).
Make sure to backup your data before you execute this command. It should work without data loss, but it's always good to be prepared.