As already mentioned, that is a very limited chip. Perhaps that was assigned on purpose.
Here is one approach: INT on RA2 is edge triggered. Use it for your signal.
Assuming the pulse idles low, set INT to a rising edge. In the interrupt, read TMR1H:L registers and start TMR1 (TICON,0 =1). Then switch to falling edge for the next interrupt and stop TMT1 (TICON,0 =0). Read your TMR1H:L registers and subtract to find out how long the pulse was.
As mentioned, if you have capture, it is easier, but for the precision you need, starting and stopping TMR1 is easy to visualize.
TMR1 has various pre-scale values, so regardless of your clock speed (within reason) you can get a count to equal microseconds or something else. If the counter rolls over just once during the measurement, that won't affect results, if your subtraction is done correctly.