The PIC Pins are Input or Output. The 8052 is pseudo bi-directional.
For the 8052 simply write a "0" to the pin, wait the amount of time the cap needs to discharge. then write a "1" to the pin and start timing.
you need a pullup for this, the internal pullup (both short strong and 'resistive') are miserably variable (chip to chip, temp, ....)
I'd use a P0 pin for this to avoid the mess.
anyhow, of youn were to use an op-amp or comparator, why not use a A/D converter instead.

Erik