Then, you can triangulate the length of the shadow, and the known height
of the dome. It's a bit uphill as you go north, I seem to recall, which
may shorten the shadow, so that should be taken into consideration.

With this, you will get the angle of the sun (declination). You now have a
known latitude, pretty close time of day, and (also pretty close) angle of
the sun.

That's all way beyond my abilities!

I could read a sundial though, I'm sure... is this one?
http://tinyurl.com/rjjs2