Replace ceiling fan with fan + light
I would open it up at the switch-box end if I were you

What I would do if I were in your shoes is open up the switch box for the switch controlling this fan to see if an extra conductor is present in the cable from the fan to the switch.  If so, then you can use a double switch to have separate wall switches for the light and fan.  If not, then you're stuck using the pull chains, a fan remote, or a specialized controller that lets you control the light and fan from a single wall device.