I'm no expert on this subject, but my understanding is that the bladder is there to protect the pump motor from starting every time you draw a glass of water from your tap. The whole idea is that the bladder provides the pressure necessary to keep water flowing in your house's water supply piping for about a minute so that your well's water pump doesn't have to keep going on and off every time there's a small demand for water. That's cuz pumps like this will draw a high current as they're starting, and that high current heats up the motor windings very quickly. Frequent starting of the pump motor can endanger the pump if it doesn't have a minute to cool down between starts, and that'll burn the motor out prematurely.
If you just pump air into the top of your bladder tank, you really have no assurance that the air is going to stay in there to provide the necessary protection. The air can dissolve in the water and you could have spurts of air coming out of the faucets in your house, and each one means there's less air in the tank to protect your pump motor.
The bottom line is that replacing the tank (if necessary) is gonna be a lot cheaper than replacing the pump in your well. I'd play it safe and replace the bladder tank BEFORE you wreck your pump. It's something that needs to get fixed, and by not fixing it you're running the risk of having to replace BOTH the tank and the pump.
Some bladder tanks allow the bladder to be replaced. Look on the tank to see if there's any way to take it apart. If not, just spring for a new tank.
Again, I'm not expert, but I'd play it safe here and replace the tank (if you can't just replace the bladder).
(I'd also like to hear what the experts in here think because I'm also wondering whether pressuring up the tank with air wouldn't make it work just like the old cushion tanks on boilers which never had a bladder.)