cp after xargs not working

While you already know how you should solve your current problem, I'll still answer about xargs.

xargs puts the string it got in the end of command, while in your case you need that string before the last argument of cp. Use -I option of xargs to construct the command. Like this:

ls /source/path/*pattern* | xargs -I{} cp -u {} /destination/path

In this example I'm using {} to as a replacement string, so the syntax looks similar to find.

ls -al /var/cache/apt/archives/ |
    grep 'i386' | 
    awk '{print $9}'

can be simplified to /var/cache/apt/archives/*i386*

So, use either of these two:

cp -u /var/cache/apt/archives/*i386* /home/alex/debian-share/apt-archives/

cp -ut /home/alex/debian-share/apt-archives/ /var/cache/apt/archives/*i386*

where

   -t, --target-directory=DIRECTORY
          copy all SOURCE arguments into DIRECTORY

See also info on parsing ls