While it might be possible to set up an NFS or SMB service and mount your disk or a portion of it remotely for the purpose you describe you may find it a lot easier to just use rsync.

if your data is on an image (I assume that’s why you say “from an old droplet”) then the answer is no, you need to use the image to create a droplet, then move the data to the new server/droplet. rsync FTW!