Things are missing, like 2 or 3 nuts that turn onto the threaded ends of the long tube. Also, there's no sign of how the lampshade is fastened. Since you have not shown a harp, I am going to assume the lampshade has two little loops at an angle that just slip over the top of the light bulb. Then I'm going to assume you should have three nuts, and that the single barrel piece in your lampparts2 photo is not a threaded item that screws on - it just has a hole in the top that lets it slide over the long tube.
So here goes. Look closely at the long tube with threaded ends. Looks like the threads on one end are shorter than the other. This is the bottom end. On it put one nut, then a small lock washer. Take the glass piece off the base, then put the end of the tube through the base from the top. Flip it over, put the other small lock washer on and then another nut. Tighten the nut so the tube is firmly mounted on the base. At this point you should have on the tube, in order from bottom to top: nut, lock washer, base, lock washer, nut.
Flip the unit upright again and slide the glass piece over the top of the tube, closed end up like in the photo. Put the large lock washer on top of that, then a third nut, and tighen the nut down GENTLY - do not turn hard enough to crack the glass! Now you have the base with a threaded tube top sticking up out of the glass. Place the wide brass bell-shaped disk over the top, then that single barrel piece (small hole at the top) over the tube. By now you should have about 1/4" of threaded tube still sticking up through the top of the barrel piece. If it's just a bit too long or too short, re-adjust where all three of the nuts on the tube are, so the top height is right.
The base of the lamp socket casing should have a threaded section and a setscrew. Loosen the setscrew, turn the base onto the top of the tube right down against the barrel piece, then tighten the setscrew. Now is when you push the end of your new cord up through the tube, from inside the base up through the top of the tube, to reach the socket. Bare the wire ends, twist and fasten into the socket electrical connection screws. NOTE: I expect this is a two-conductor cord, and you say the original was not polarized. Today even two-wire cords (with no ground) can be polarized - on the plug, the wider blade is Neutral, and the narrower one is Hot. If you have that plug style, the HOT should connect to the light socket's center contact, and Neutral to the screw shell of the socket. With a three-way socket, you still make sure the Neutral line (wide plug blade) connects to the socket's shell.
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