Hi,
I'm running Mint Mate 18.2 and have a Fritz!Box 7360.

What I'm trying to achieve is to get DDNS with No-IP running.
dynamic DNS is set up with the NO-IP tool for Linux
I'm lost when it comes to network settings.
My portforwarding on Fritz!Box is set to:

When I now try http://www.portchecktool.com/ to see if any of my 80... ports would be open I just get I could not see your service on IP address on port (88).
I have no firewall running.

Can you please point me into the right direction?
I hope I posted all the information which is needed.
thanks

Welcome to LinuxQuestions.

Basically the purpose of a no-ip is to track your dynamic IP address which will automatically resolve when you use whatever URL you selected. The no-ip client can run on any computer or possibly your router. If you ping your URL does it show the correct IP address?

Port forwarding allows someone from outside your internal network to access some service running on your LAN. For example a web server by default uses port 80. Normally you would add forward port rule on the router from port 80 TCP to your server port 80. Also if a firewall is running (not in your case) on your server you need to allow incoming connections to the web server or port 80. Someone from outside could then access your webserver by using your no-ip URL.

While the from port can be almost any valid port the to port needs to be an actual service running using that desired port. You picked from 8080-8084 to 83-87. There is no known service that uses 83-87 by default so there is nothing for portcheck.com to "see".

An easy service to check is ssh which uses port 22.
Start ssh if not already running.
Add a rule to forward port 22 TCP to your server port 22. (Not sure if the router has internal DNS capability so I would use the computer's IP address and use a DHCP IP reservation to make it "static".)

Now you should be able to access the server using
ssh myuser@my.no-ip.URL
Check port 22 using portcheck.com