RolaniXor's suggestion plus a couple additional steps is what ultimately worked for me. I've seen a few of these threads and none that are solved, so please share if you can confirm the following experience is a common solution.
Note: This machine is running a 64-bit lxde desktop.
About a week ago, an upgrade was performed from 11.10 to 12.04 and received the same "Partial Upgrade" popup in the Update Manager. This was ignored after reading a lot of problems people have after running a partial upgrade. I continued to update whatever packages were available via Update Manager.
Today, I saw some kernel packages as well as the sessioninstaller package which I was not able to update through Update Manager. These were successfully added using sudo apt-get install from command line. The package called sessioninstaller was the one that ultimately stopped the "Partial Upgrade" suggestions and allowed me to update other packages which were previously disabled via Update Manager.
This may or may not be the case for others, but if you see sessioninstaller and you have updated all available packages in your Update Manager, you may wish to try installing it from command line with: sudo apt-get install sessionmanager
Here are my update logs containing the kernel packages as well, although I don't know enough to know whether or not that helped with this situation. I just figured kernel comes before distro and installed these packages first. See below.
Start-Date: 2013-05-16 18:49:21
Commandline: aptdaemon role='role-commit-packages' sender=':1.64'
Upgrade: php5:amd64 (5.4.14-1~precise+1, 5.4.15-1~precise+1), thunderbird:amd64 (17.0.5+build1-0ubuntu0.12.04.1, 17.0.6+build1-0ubuntu0.12.04.1), firefox-globalmenu:amd64 (20.0+build1-0ubuntu0.12.04.3, 21.0+build2-0ubuntu0.12.04.3), libapache2-mod-php5:amd64 (5.4.14-1~precise+1, 5.4.15-1~precise+1), mtools:amd64 (4.0.12-1, 4.0.12-1ubuntu0.12.04.1), php5-gd:amd64 (5.4.14-1~precise+1, 5.4.15-1~precise+1), firefox-branding:amd64 (20.0+build1-0ubuntu0.12.04.3, 21.0+build2-0ubuntu0.12.04.3), lib32asound2:amd64 (1.0.25-1ubuntu10.1, 1.0.25-1ubuntu10.2), firefox:amd64 (20.0+build1-0ubuntu0.12.04.3, 21.0+build2-0ubuntu0.12.04.3), alsa-utils:amd64 (1.0.25-1ubuntu5, 1.0.25-1ubuntu5.2), libasound2:amd64 (1.0.25-1ubuntu10.1, 1.0.25-1ubuntu10.2), libasound2:i386 (1.0.25-1ubuntu10.1, 1.0.25-1ubuntu10.2), firefox-locale-en:amd64 (20.0+build1-0ubuntu0.12.04.3, 21.0+build2-0ubuntu0.12.04.3), flashplugin-installer:amd64 (11.2.202.280ubuntu0.12.04.1, 11.2.202.285ubuntu0.12.04.1), php5-mcrypt:amd64 (5.4.14-1~precise+1, 5.4.15-1~precise+1), steam-launcher:amd64 (1.0.0.38, 1.0.0.39), thunderbird-globalmenu:amd64 (17.0.5+build1-0ubuntu0.12.04.1, 17.0.6+build1-0ubuntu0.12.04.1), php5-mysql:amd64 (5.4.14-1~precise+1, 5.4.15-1~precise+1), php5-cli:amd64 (5.4.14-1~precise+1, 5.4.15-1~precise+1), php5-common:amd64 (5.4.14-1~precise+1, 5.4.15-1~precise+1)
End-Date: 2013-05-16 18:51:22
Start-Date: 2013-05-16 18:53:30
Commandline: aptdaemon role='role-commit-packages' sender=':1.64'
Upgrade: linux-libc-dev:amd64 (3.2.0-41.66, 3.2.0-43.68)
End-Date: 2013-05-16 18:53:34
Start-Date: 2013-05-16 19:02:21
Commandline: apt-get install linux-headers-generic
Install: linux-headers-3.2.0-43:amd64 (3.2.0-43.68, automatic), linux-headers-3.2.0-43-generic:amd64 (3.2.0-43.68, automatic)
Upgrade: linux-headers-generic:amd64 (3.2.0.41.49, 3.2.0.43.51)
End-Date: 2013-05-16 19:02:48
Start-Date: 2013-05-16 19:03:57
Commandline: apt-get install linux-image-generic
Install: linux-image-3.2.0-43-generic:amd64 (3.2.0-43.68)
Upgrade: linux-image-generic:amd64 (3.2.0.41.49, 3.2.0.43.51)
End-Date: 2013-05-16 19:04:43
Start-Date: 2013-05-16 19:11:07
Commandline: apt-get install sessioninstaller
Upgrade: sessioninstaller:amd64 (0.20+bzr120-0ubuntu2, 0.20+bzr128-0ubuntu1.2)
Remove: gnome-codec-install:amd64 (0.4.7+nmu1ubuntu3)
End-Date: 2013-05-16 19:11:47