James gave you the correct procedure, you just need to adapt it to your situation. Since your old / is mounted at /mnt rather than /mnt/sysimage, you'll want to get (cp) the rpm you're trying to install some place under /mnt, probably /mnt/tmp.
For each of the partitions listed in /mnt/etc/fstab, you'll need to get them mounted in the right spot under /mnt. For example, if you have /usr and /var as separate partitions, you'll need to
mount /dev/<h or s>d<whatever> /mnt/usr mount /dev/<h or s>d<whatever> /mnt/var
You fill in the device files based on what they are in /mnt/etc/fstab. If the device files aren't listed (you're mounting by LABEL=) you need to use `tune2fs -l /dev/<h or s>d<whatever>' for each and every partition you have, until you find the one that corresponds to the right label, and then mount it in the right spot under /mnt.
Once you have the necessary partitions mounted, so that your entire system is under /mnt in the right places, you
chroot /mnt
and then install/upgrade the RPM that's sitting in your new /tmp
rpm -Uvh /tmp/your-rpm.rpm
Tim