The Nano can be somewhat useful as a rudimentary single channel logic analyzer (especially if nothing else is within reach), but it is not the right device to use or build on for this purpose. It can however add to the logic analyzer and show you the true nature of the input signal. This can help you identify noise, ripple, transients etc. It will also allow you to do a quick verification of signal frequency (do I have the right baudrate/speed) and also check for presence of clock and other digital bus signals.
For capturing and analyzing long streams of logic however, it is totally inadequate.