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As another way to use rising and setting of stars -- how about
using the compass to identify the azimuth at which some circumpolar
stars rise and set?  I suspect that could tell you your latitude
with reasonable accuracy, even without a watch.  Even if you can
only read the compass to a couple of degrees, I think the
uncertainty in such a latitude will depend more on the uncertainty
of refraction at the horizon.  For example, consider a star whose
path just grazes the northern horizon -- that tells us our latitude
without being very dependent on the azimuth measurement.

With a watch, you can get latitude from the length of time a star
far from the equator is above (or below) the horizon.  Perhaps you
could also estimate when it's directly in line with the celestial
pole, allowing you to measure some stars whose rise or set occurs
during the day.
        -- Bill

   

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