How did you try?
For a couple of bigger puppets I was able to stick fur fabric over the main part of the body. But it didn't look good for skinnier parts like the limbs, it made them like fat sausages or those fur mufflers, 3 times the diameter of the actual limb and with no difference between the big muscles and the thin ankles. Some fur fabric has a stretchy backing, and that worked best on the body. Some does not stretch. But either way, it adds thickness, and the better quality fur fabrics have a wooly undercoat of shorter fur as well as the longer hairs, and that bulks it out. If you want shorter fur going down the legs or on the nose, you can try trimming the fur, but it ends up looking wooly and not like the fine straight hairs you would see on a cat's nose. You want shorter hair, but more like the long hairs on the surface of the fabric, not the wooly undercoat.
One way is to cut rows of hairs off the fabric with scissors, and glue them onto the puppet. That avoids the thick backing, and lets the puppet bend better. If the hair is pointing to the back of the animal, you start applying the rows of fur at the back, with it pointing back a bit, then the next row in front of it. I think some people use latex as a glue, but it is slow drying and doesn't grip when it is wet. I have used upholstery adhesive, which is a contact glue in a spray can, like 3M No. 74 Foam and Fabric glue, which might be called Foamfast now. But spraying is hind of messy since you don't want to spray all over the hairs you have already stuck on, so contact adhesive you brush on would be easier to work with and cheaper.
I also use flocking for the shorter hairs, but you need an electrostatic flocking gun for that. And it only does short hairs, it doesn't give you the full range of lengths. If you just sprinkle it over the glued puppet, it lands at all angles, the gun charges it so it magnetically attracted and lands end on.
To be honest, the last time I had a design for a hairy monster for an indy feature film, we changed the design so it had bare skin, because none of the hair tests were looking good enough.