So I have this old wall lamp with a different color code that I'm used to: There are 2 brown wires welded together and a separate dark brown wire. As seen on the picture the 2 brown wires leads to the light bulbs and at the connector / switch it says "2A / 250V" meanwhile at the dark brown wire it says "A1 / 31". The thread on the picture is the switch, to be pulled up and down. I'm not sure how to go about it and I rather not risk it so I turn to your wisdom instead!
Unscrew the bulbs and see which wire is connected to the side-contact that contacts the thread of the bulb using a continuity tester. That one is the wire you want to be the neutral.
That way the live contact is buried deep within the socket and harder to reach by inquisitive fingers.
The dark brown looks to be the ground connection. Check that it has continuity to the metal housing to verify.
The two brown wires "welded" together appear to be the neutrals and should be hooked to the white wire or neutral. You could /should verify this as ratchet freak suggests. The single brown wire appears to come from the switch and should be the feed and should be hooked to the black, hot, wire. You always want to switch the hot feed. The other two brown wires appear to go into the load side of the switch.
Installing this will be tricky. It's meant to be installed hanging on a wall because of the keyhole notch at the top. There should be some sort of bracket/cover that goes over the back to cover the wires and the feed, black and white wires, would come from a properly mounted box on the wall. It doesn't look like it was meant to have a cord go out the bottom and plug into an outlet along with the pull string, but could be done this way.
There isn't a ground wire that I can see but one could be added.
For a lamp, you want the neutral to connect to the outer threaded part of the bulb socket. The switched hot wires should connect to the center contact in the bottom of the socket.
You want the switch on the live wire so when it is switched off there will be less potential for something to go wrong. Cut the messy end off the dark brown wire so that it can be stripped back neatly and connect it to the live feed.
Mains wires shouldn't really be soldered together. If the connector in the wall gives you the opportunity, you could cut off the soldered joint and fix each wire into its own connector - or use a Wago connector for stranded wire which has enough holes.
Ideally, get a qualified electrician to install it for you. I am not a qualified electrician.