Applying a texture to part of a sphere

A quick and dirty way to do this is to pad the image with pixels of some inoffensive color, so that the actual meaningful part of the image gets mapped to a smaller portion of the sphere:

image = Import["ExampleData/ocelot.jpg"];
{width, height} = ImageMeasurements[image, "Dimensions"];
image = ImagePad[image, {{2 width, 2 width}, {height, height}}, White];
SphericalPlot3D[1, {Theta, 0, Pi}, {Phi, 0, 2 Pi}, PlotStyle -> Texture[image], 
  TextureCoordinateFunction -> ({#5, 1 - #4} &), ViewPoint -> Left]

The first three lines create a new version of your image that's padded with white pixels on all sides; you'll need to play around with your values to get the aspect ratio right. Projecting this image onto the sphere then yields the following:

Note that I changed the ViewPoint in order to get the image on the "front" of the sphere. With the default viewpoint, you end up with the image about 3/4 of the way to the other side of the sphere.

The other way to do this would be to set TextureCoordinateScaling -> False and then rescale your TextureCoordinateFunction (put numbers in front of #5 and #4 and see what happens.) But I'm not sure if there's a way to do this without having the image repeat itself ad infinitum.

You can achieve this using "Texture". However, to prevent that the whole sphere is covered with the text, you must pad the text with White:

ima = Graphics@Text[Style["Hello", FontSize -> 100]];
{width, height} = ImageMeasurements[ima, "Dimensions"][[;; 2]]
te = ImagePad[ima, {0.5 {width, width}, 0.1 {height, height}}, White];

ParametricPlot3D[{Sqrt[1 - z^2] Sin[ph], Sqrt[1 - z^2] Cos[ph], 
  z}, {ph, 0, 2 Pi}, {z, -1, 1}, 
 TextureCoordinateFunction -> ({-#4, #5} &), 
 PlotStyle -> Directive[Texture[te]], Mesh -> None]