Everything went fine, except for one thing: while booting, I noticed a message I had never seen before. It was something like:
A start job is running for dev-disk-by (a long string of numbers here,
followed by the time count: x/1min30sec)
Fortunately, it actually lasted the one minute with 30 seconds it announced, but that made my boot time longer than 2 minutes!
Thanks to this Arch Linux forum, I could find the answer. Basically, I had to edit the etc/fstab file to make it accurately point to my swap partition. In my case, /dev/sda2.
That took care of the issue and now my Pardus system boots normally.
Oh, and the upgrade seems to have fixed Steam. Now I can play Violett again! :P