Of course, the image needs to be big for this, but the plugin you would use is CC Sphere.
Drop your movie into a composition and apply CC Sphere to the video.
Then set the Render option in the effect to "Inside".
Under Shading set Ambient to 100% and Diffuse to 0%.
Then scale inside the effect until the image fills the screen.
Then you'll be able to use the Rotate Y controls in CC Sphere to spin around inside your video!
You can also add Optics Compensation and Reverse the Lens Distortion to straighten out your image.
Let me know if you have any questions!
AE will not output 360º videos. That requires other software unless you are talking about mapping the images from 6 cameras to a sphere and then rotating around the sphere in AE. The biggest problem I have directing you down the right path is that I don't know what you are trying to achieve. If you art trying to make one of those videos that you see on the net where you can look around in any direction as the video is playing then AE is not going to be of any help except for possibly stabilizing or color correction on the 6 original streams of video. There is no way for AE to render a VR video or a VR still image. You have to look elsewhere.
I am working with a video that has already been stitched together and am looking to rotate it within a sphere. Basically, I would like to adjust the camera as the 360 video plays within AE to show a demo of what it would be like to move the camera as you see in the other 360 players. I am not looking for the same functionality as a 360 player, just the appearance of functionality.
If you are video has been stitched together so that it looks like a rectangular flat projection then all you have to do is make a comp from that video and then mess that comp in your main comp and apply CC Sphere to your nested comp and expand the radius so you are inside the sphere.
If your stitched together are not a rectangular projection then you will have to do something to make it so. Here is an example of what a rectangular projection would look like.