Which iOS app will let me paste an image?
Since you're able to copy the image, open up the Messages-app, open a new message to your own iMessage-address, and paste, then send. You should be able to save the image to the camera roll from there.

On newer iOS like 10 and higher, the ubiquitous share sheet function lets you move documents around so much easier, that third party apps aren't needed for this basic function anymore.