Yep. Switch supplies/cuts power to the whole light/fan combo and the pull strings on the light/fan control the fan controls or whether the light is on/off while the fan is running.

Or you can use a remote to control the light/fan.

Yes, most ceiling fans either have pull chains or a remote control available to allow you to switch the fan and light features separately from one power feed.

The bigger issue is making sure you have adequate support for the fan. If the box was designed for a light only, it probably isn't sufficiently strong to support a fan. Of course, some houses are roughed in for fans to start with even if they only put a light in. If it's not a fan box, you can retrofit a fan box. They make ones that you put in the support bar through a normal box sized hole. No additional damage to the wallboard/plaster requried.