there normally will be two ways to do this

the first is to "bake" a texture using "blender render"
the second is using the camera and nodes in "cycles render"

as in the old way VS the new way


i am assuming you can switch from blender render to Cycles Render and back


i am also guessing you are by default in blender render

so for that you need to make a UV sphere just slightly bigger than the mesh you have
uv map it to spherical projection
in the outliner window ( top right corner ) click on your world
hold down < control> and click on the new uv sphere

then under bake select "texture" then click the bake button



for cycles and nodes ( better results )

move blender into cycles render
check " use nodes" box in miterales to creat a new node set up
add a new image texture node and in the drop down menu should be the textues name you are using for the oddly mapped textuer

add a emission shader ( this will cause the texture to glow)

add a camera at the center and set it to equal area projection
( you might need to rotate X + 90 degrees )
--- sounds very complex but really it is not
and render


it might be easiest if you can pack the texture into the "*.blend" file and post it

-- with blender running and everything looking good select from "file" ( top left ) "external data / pack all into blend "
then post the blend file ( might have to zip it for some services to except it )
i know that my FREE google storage ( from using gmail) will handle a blender blend file just fine

this way i can set things up for you and you can then study what i post back