Boot your machine from an Ubuntu live CD/DVD or live USB stick (like you would use to install Ubuntu). Select
Try Ubuntu (not
Install Ubuntu). Run the GParted Partition Editor (you can search for
gparted in the dash).
Shrink down your 581 GB partition (
/dev/sda6) to make room to the left. Shrink it by however much space you want to add to the
/boot partition--in this case, by about 408 MB. (You may have to shrink it, then move it to the right.)
Then expand the
/dev/sda1 partition to take up the newly available unpartitioned space.
Apply your changes by clicking the green check mark.
Quit GParted and reboot back into the Ubuntu system installed on your hard disk. Now your
/boot partition is about 500 MB in size, and your
/ partition is shrunk by about 408 MB to make room for it.
Finally, please note that you should not attempt to change the sizes of partitions on the system drive from the installed Ubuntu system. At best GParted will refuse to operate and at worst you can severely mess up your Ubuntu system and even have data loss. This is why you need to boot from a live CD to perform these operations. That is relatively safe, but whenever you dynamically resize partitions there's a small risk of data loss (either if the operation fails or if you make a mistake), so you should make sure all your backups of your documents and other important (i.e., irreplaceable, or tough to replace) files in your Ubuntu system are current before you perform the above procedure.