Just wanted to say thanks to all! I was able to get it done. To put the cups into the frame I used a c-clamp with some blocks of wood.

I had a small issue in that I had some extra cups that seemed unnecessary. I don't know where they came from. They did fit together with the other parts, but when everything was tightened it was impossible to turn the crank. I removed them based on a youtube video where he didn't have those, and then it worked fine. I have no clue what they were! (They are the small round ones in the 3rd pic of my first post.)

All in all it was kind of a pain (especially getting the cups into the frame) but I'm so happy I did it!