Ipad still sees wifi-network but no internet connection

If not I might try a wifi repeater to extend my main wifi network.

Assuming this is not an apple router it could be a good idea.

The airport extreme gen1 is ancient.. I mean old.. like Noah using it on the Ark.

The Gen1 AE is not dual band simultaneous so it could just be extremely poor wireless signal or other faults in a unit that is now 11 or more years old.. time for update methinks.

But when trying to use my iPad or iPhone in the Patio that is next to the basement I get dropouts multiple times a day.

As a test, what happens if you power off both the iPhone and iPad and move them to the same location as the Mac mini and power them back up again?

If the iPhone and iPad stay connected during your "test", then this tells you that they are not picking up a strong enough signal when they are located outside on the patio. There can be several reasons for this, but the usual reason is that the outside walls in your home are too dense to allow decent WiFi signal quality outside on the patio.

Another thought would be that a 1st Generation AirPort Extreme is at least 10 years old and as much as 12+ depending on when it placed into service. The average useful life of this product was about 5 years, so it is remarkable the product is still functioning at all.

Even though the AirPort Extreme is still working, the power supply and WiFi radio have likely been heading downhill for some time, with the result being that the AirPort does not broadcast nearly as strong a WiFi signal as it did even a few years ago. The signal might be good enough where the Mac mini is located, but too weak if the signal must pass through obstructions to reach the patio area.

Apple classifies the 1st Gen AirPort Extreme as "Obsolete" meaning that they will not help you with the product, even if you are willing to pay them.

Unfortunately, the iPad and iPad are poor diagnostic tools as far as WiFi is concerned. In order to run a few more meaningful checks, it would be very useful if you had a Mac laptop that you could use to run some additional WiFi signal quality checks. Is that a possibility?

Would it maybe help to fiddle with the DNS or IP dress settings on the iPad?

No, if the issue is WiFi signal strength and quality at the patio location, which is what I suspect. But, we'll know more after your "tests".