Right all you need is 1GB of space to get a day (you say that in a comment)?
Easy way (i think) is to clear your old logs, if they are not important by:
Or just moving your old logs to another place
mount /dev/sdc1 /mnt/pendrive
mv *.gz /mnt/pendrive
mv *.gz /mnt/pendrive/myfunnyapache2folder
Or if for some unknow reason, you dont want to do it, we can try moving /var/log to another partition until you get everything fine!
First stop your services: service rsyslog stop; service apache2 stop ; service mysql stop, etc
fdisk /dev/sda (change sda to your disk device or pendrive)
n (using n command create your 1GB partition and save fdisk with w command, and just quit fdisk)
Check with fdisk -l (L lowercase) what partition device your 1gb partition assumed, for example /dev/sda5
Now, mount /dev/sda5 or your partition number (created by fdisk) as your new /var/log...
mkfs.ext4 /dev/sda5 - Assumed /dev/sda5 as the new empty partition. This will destroy all that in /dev/sda5 if you select the wrong partition, this is the empty partition you created in fdisk!
mount /dev/sda5 /tmp/log
mv /var/log/* /tmp/log
mount /dev/sda5 /var/log
ls /var/log - Now you have time to think again...
After that you will be able to restart your services: service rsyslog start; service apache2 start; service mysql start, etc...
Yes it can be dangerous if you dont pay attention in what you are doing, but less than trying to resize or merge partitions on a running system.
If you really dont have any more space on your hard drive, create a ramdisk partition (you can found how to do that in google), but you will lost your log data if the server restarts....
You can compress non-rotated log data too, with gzip, like that gzip filename, and other text files, that you probably will not need right now!