Since I'm quite new with electronics I would need some advice for powering an arduino nano from a wall socket.
I've built an alarm clock using the Arduino nano ATmega328p, for the wiring see the attached image. It works great when powered by the usb-c from my computer. However, I've tried using a phone charger (with max 0.5A 4V) to power it through the usb-c, but then the chip just exploded. I thought that the specification on the phone charger defines its maximum outlet (here 0.5A 4V).
Any advice on how I can power it from the wall socket?
Powering using a phone charger into the USB socket should work fine, so something isn't as you think it is. Please post some photos of what you have, making sure that the photos are clear. A voltage above 5V would cause the damage you described, so perhaps the power supply you are using might have an output higher than 5V.
The missing resistor in series with your LED might also cause damage, perhaps use 220 Ohms.
A 5V 0.4A phone charger should have been absolutely fine for your project.
But you also mentioned USB-c. Unlike older technologies, USB-c is capable of delivering higher voltages than 5V, like 12V or 19V, in order to deliver higher power without higher currents. Those higher voltages could in theory cause the problems you described. But this should never happen by accident. USB-c chargers are "intelligent". Those higher voltages are only ever supplied on request, and the request mechanism is pretty secure, higher voltages should never be delivered unless requested in a very particular way, with the charger and device digitally negotiating it.
You also mentioned using a Nano. At one time, a Nano was a Nano, meaning a board very similar to an Uno R3 but with an different form factor. These days, there are many variants of both Uno and Nano with very different specifications, so please post a link to the board you have. Maybe it is not the atmega328 based board that you think it is? (Although I have never heard of any board called a Nano that can request >5V from a USB-c charger.)
Beware! When we got new cell phones, we also got a "fast charger". Looking at the writing on it I see it can provide up to 9 volts or up to 11 volts for charging. I am sure all that is assuming communication with a cell phone, not connecting to an Arduino. But I would not use the thing to power any project.
