Hi,
Since your mixer doesn't have USB connectivity, you can use any external sound card to record your mixes. As you mentioned, the Behringer UCA-222 (or the UCA-202), or any other sound card which have line level inputs* (this is important to handle the right intensity), should be enough to record the master output or the REC OUT. I do recommend you to use the REC OUT, if available, instead of the master out.
You will also need a PC/MAC with the sound card connected, plus a software like Audacity or Soundforge to record the mix.
Some people also buy 1/8 to RCA cables, to record from mixer to line in/mic into the PC or recorder, but from my experience, I recommend you to record with a proper RCA cable and sound card, or directly from your mixer using a USB cable (in mixers such as DJM-450, 750, 850, 900).