I imagine upgrading your digital ocean server plan isn't as easy as the click of a few buttons. I assume I will have to migrate my drupal website to the new server as well?
You can easily do a fast-resize from the control panel and this only requires a reboot. The fast-resize will increase the RAM and CPU for your droplet but will not resize your disk image. If you need to increase your disk size as well you can create a snapshot of your droplet and use that to re-deploy to a larger size. If you delete your current droplet just before creating your new one (in the same region) the new droplet will inherit the recently destroyed one's IP address as well.