How to measure pulse width of an IR signal using an 8-bit PIC?
Yes, the TSOP1738 will do at this short distance. The 0.65 relative responsitivity means that at 36 kHz your IR LED needs to be \$\sqrt{0.65}\$ = 0.8 times closer to see the same signal strength, due to the inverse-square law. So if your TSOP1738 sees a certain level for 38 kHz at 1 m, you'll have to hold the transmitter at 80 cm to get the same signal strength at 36 kHz. BTW, with a remote control with fresh batteries I measured perfect reception at more than 15 m distance, so no problem at all.

Don't worry about the PIC's performance. The TSOP1738 won't output the 38 kHz signal. That's the carrier frequency, which is removed by the TSOP1738 to get back the baseband signal, which has a much lower frequency, with pulse durations in the order of 1 ms, so there's plenty of time to measure time between edges accurately.  

The following scope images illustrate this:  

  

This is one RC5 code. The top signal is the 36 kHz modulated signal, the bottom the baseband signal with the actual code.   



This is zoomed in on one pulse of the baseband signal. You can see individual pulses of the 36 kHz carrier.

One more word about the carrier frequency. You may be using a remote control which you don't know this frequency of. The TSOP1738 doesn't give it on its output, so if you want to read it you'll have to connect an IR photodiode or transistor to one of the PIC's inputs and read the time between two same edges. That's feasible. Period times for different carrier frequencies:  

40 kHz: 25 &micro;s
38 kHz: 26.3 &micro;s
36 kHz: 27.8 &micro;s  

A 20 MHz PIC16F616 has an instruction cycle of 200 ns (it divides the clock by 4!). So readings for the three frequencies should be about 125, 131 and 139. That should be enough to tell them apart. But if you want you can let a number of edges pass and only read the timer after the 10th interrupt, for instance: 1250, 1316, 1389. Not too much longer because you have to keep the time shorter than one pulse of the baseband signal.

Success!