Article: Screw a flexible hose on the end (for pumping up bike tires) and screw the hose onto a bike pump.  Note that you can seal the hole in the ball with strong glue if it tends to automatically re-inflate itself, then just re-pierce the hole with a skewer once the ball is in the bottle.
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Puncture the ball in the fuzzy area (part the fuzz so that the hole doesn't rip out too much of the fuzz, you will comb it back over the hole later). Put the ball in a vice and squeeze all the air out of it. Seal the hole with a bike pump needle. Fold the ball up (or roll it) and poke it into the bottle. Turn the bottle upside down so the needle pokes out the neck. Re-inflate the ball and then pull the needle out. Disguise the hole by combing over the fuzz and/or putting a drop or two of paint over the hole.