First of all, thank you for your fast help.
I tried both solutions, but unfortunately they didn’t work. I set the display to never turn of, wiped Cache/Dalvik and started the update again. I left the phone active overnight and at the next mornig, it still needs "Just a sec…"
Any new suggestions?
Just to avoid confusion (as I think your post may have been mistakenly moved here): The screens you describe, do they occur when you try to log in to the phone for the first time after the update?
Also, which version were you on before the update?
The screens I described in the prior post came after I started the Fairphone Updater, downloaded the update, waited for installation and then restarting the phone (as required).
Sorry, which version it was before I do not know. Right now it has the FP2-userdebug 5.1 FP2 fp2-sibon-16.06.0 Version.
I created a backup with TWRP and resetted the phone to factory settings. After booting, the “Just a sec…” Screen disappeared in seconds and I was able to use the phone as usual (of course without my data on it).
However, with my backup recovered, it is just like before. It has to be something connected to my apps, I suppose…
Well, that’s not surprising, as the TWRP backup makes an exact copy of your device’s file system. So whatever went wrong before is also safely stored and preserved for future generations on this backup 
Thus, if you want to backup data, settings, and apps on a software-broken phone, I would recommend to also (or only) backup data manually, and/or apps and settings via titanium (or something similar).
I pointed you to a post named “How to update a rooted FP2 with TWRP”. Within this post I have written about a backup, but in your action I miss “update”
@Spielmops I first tried the backup way because in the instructions you linked, one had to to downloads with the phone. However the “Just a sec…” Screen prevents me to start any app (for instance browser) and therefore I couldn’t download anything. Also I was not sure if this way would erase my data or not. Maybe you can answer this question?
You got me: I forgot to mention, that the update-procedure does not affect your data. Will update that post later. 
It’s easier to do the whole action incl. downloads from the phone, but if those downloads are not possible with the phone, you can gather the needed datas per PC and store them on sdcard.
@Spielmops I have a question on step 2 ("Download the new boot-image with root from here: http://siteactif.org/fp2/15.")
How do I know which file to download?
Thank you in advance
It worked! Thank you very much!! And all my data is still there. @Spielmops, you are my hero of the day 
One last question: It seems like I now have the standard version with google. Is there a version for the non-google version or a way to change it?
Yes.[quote=“Phoenix2222, post:74, topic:16372”]
a way to change it
[/quote]
Yes
Search here for your Live without Google …
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