Well, you might have to install Fairphone OS first if you haven’t got it installed yet. I guess Fairphone’s storage update might not work with stock Android. @paulakreuzer ?
Also be prepared that you might have to install (Fairphone OS) manually, i.e. from your SD card in recovery mode.
Okay, the Fairphone Updater says that my current version is Fairphone Kola Nut 1.8.7. So I must have Fairphone OS, right? If yes, there must be another problem. Supposedly, my storage is running out and some functions are not working correctly anymore …
Since you didn’t know you had Fairphone OS 1.8.7 installed I have to ask: Did you use the right storage upgrader?
Sorry for the confusion, yes, this was exactly the link I used. I tried several times …
Sorry if this bugs you, but just to avoid any misunderstanding: When you say you tried updating manually, you went through recovery mode and applied the storage upgrade file (while being in recovery mode) that was saved on your SD card?
I sourced this from another thread:
@Barth means the “wipe cache partition” option in recovery mode.
No problem. Yes, that‘s what I did. I followed every step of the tutorial.
Thanks Stefan, I also found this thread about clearing the cache before rebooting, but this also did‘t work. Hmpf! 
Since the Unified Storage Upgrader deletes all files on your phone anyways, you might as well try wiping data before applying the Upgrader. Does this help?
Thx, I‘ll try this in the next days and tell you if it worked. 
I tried everything that is adviced to get get one partition for koala nut, but still have two partitions so cannot install the required update for Whatsapp because there is not enough space in internal storage (there is 101 MB free, I thought this should be enough). Is it advisable to go back to factory settings ?
The Unified Storage Upgraded wipes all your data anyways, so yes, you could try a factory reset.
PS: It is Kola Nut, not Koala Nut. 
Stefan thank you for your help. Kola or Koala my FP1 seems ‘nuts’ 
: even factory reset doesn’t work. Possibly the reason why the unified storageprogram also doesn’t work. Hope for a solution and can be happy again with my FP
Did you already try to downgrade to Cherry 1.6 and do the old unified storage upgrade?
Thank you Irena. Yes, but also downgrading did not succeed, possibly may for the same reason: there is no reset. When I try a lot happens, but at the end nothing has changed.
It’s holidays, so I assume you have some time to try the following: Install ADB and sideload the *.zip from your computer through Recovery Mode. This is what I’d try if everything else fails. 
Well, there’s also the procedure to recover from a hard-brick, though it that should really be a last resort.
Hi Irina,
I downgraded to Cherry 1.6, but in the Fairphone Updater no „Storage Upgrader Cherry 1.6“ appears. Do you have any other tip before I go back to Kola Nut 1.8.7?
I think you can only install the Storage Upgrade for 1.6 (!) manually now. It’s still available for download at http://storage.googleapis.com/update-v1_6/FP1_Fairphone_OS_1G13G_partitionUpgrade_v1_6_OTA_2014-07-31.zip
Thank you, Urs! It finally worked after I changed the settings of my computer: Safari always unpacked the zip file, so that I didn‘t use it properly for the manual upgrade. Now it‘s done. Probably it would also have worked with the Kola Nut version. Thanx to all! 