How can I turn it on without using the power button

On most of those computers, if you interrupt the power by pulling the cord and removing the battery, the computer will do a restart when the power and/or battery is replaced. That may get it going again. However, I have a feeling that your problem may not be the power button. Did you press and hold it for a few seconds?

Also, you could have had a bad battery for a time and then had a connection problem with the power cord. This could deceive you into believing the on-button was bad. Just some things to consider. The Acer may be old enough to have a bad battery by now.