NavList:

Joshua
I’ve got to be quick, because I’ve more than used up my NavList time ration for this week and probably next week too.  I was about to ask Frank to ban me for a fortnight. 
First, give up thinking about assumed positions, or even fixes for a moment.  I hope Frank won’t mind my saying that his puzzles are mainly brainteasers sparked by something he’s noticed in the press or on film in the last couple of days.  The puzzle I set myself was to test a possible solution I hoped might work with the puzzle Frank set, but on my own ground.  The hope was that if the problem I set worked for me and others, it was worth trying to solve the puzzle Frank set.
 
Re: “like an exam question”.  Correct, you hit the nail right on the head.  This was mainly just a fun puzzle for readers to exercise their overall knowledge, experience, and the bits they’ve not used for a while.  It’s not that relevant to everyday practical navigation.
 
Next point.  Without date and time, there’s no way you could get a fix from the information given.  However, like with Polaris at any time, and the Sun at noon, you can get a latitude.  The reason is because the Earth and the Celestial Sphere have the same spin axes.  Imagine a grapefruit sized Earth.  Around the grapefruit is a much larger hollow Perspex sphere with the positions of the stars marked on it.  This is the Celestial Sphere.  In fact, despite what we see when we look up, it’s the Celestial Sphere that’s fixed in space and the Earth spins that round inside it.  Therefore, if you were to drill a small hole in the sphere with a Perspex drill at the position of Procyon and poke a 2B pencil through and then spun the grapefruit, you’d draw a circle at a latitude equal declination Procyon around the grapefruit.  The reason we don’t see the stars going horizontally across the sky or going up and down vertically is because we’re neither at the Pole nor the Equator.  At any other latitude on the Earth, the stars appear to circle Polaris.

Pressing on, let’s mark the observer’s zenith on our Perspex sphere.  From the positions of Procyon and Rigel scratch an arc equal to the two zenith distances (90-Ho) measured by the observer.  The arcs will cross in two places.  Take up your Perspex drill again; drill holes where the arcs cross; and stick two pencils in.  Spin the grapefruit and you’ll get the two required latitude lines marked right round the grapefruit.

So, what about a fix?  There isn’t one.  That’s the catch.  Instead, Frank tells us our longitude is 165E.  He also provides a picture of a church surrounded by palm trees.  Hopefully, you’ll find one fix in an area where palm trees grow and the other where the locals have icicles hanging from their noses.  And that’s it.  You know where you are.
DaveP