NavList:

Let me propose a solution of Kelvin's problems.

The main part, of course is how to measure any angles with a rope alone. I propose this: find two nearby stars, compute their distance from amanac, then stretch the rope, and rerord this dustane on the rope (with two knots), as seen when the rope is at your stretched arm distance. This gives you a unit of measure on the rope, with which you can measure moderate distances.  For larger distances this becomes unprecise (since you are measuring chord, whose radius is your arm). You can take this into account, by measuring the length of your arm (more precisely the distance from your eye to the arm end), using the same rope, If an assistant is available, or perhaps some convenient tree you may fix your stretched rope at a greater distance from the eye, which improves accuracy.

Kelvin took are to include a "light to read the almanac". But he did not include a pencil to do calculations!! (Paper is not so necessary since one can use the blanc pages and/or margins of the almanac. A flat ground covered with fine sand can in principle replace a pencil.

I estimate that the proposed method with extreme care and dexterity can give maximum 1 degree accuracy in measuring the angles.
This is 30 degrees error in Longitude (2 time zones!)

So your ability to tell Altantic from Pacific ocean depends on where exactly you are. I suppose it is hopeless if you are near Panama:-) But continental USA has width approx 4 time zones, so in these latitudes you may succeed.

Determining day of the month is not difficult if you have almanac with Lunar distances. In principle, even phase of the Moon can be used, though you can hardly tell the day just by looking at the Moon:-)

I also briefly looked at other exams. It is amazing what amount of knowledge was required! One was supposed to be completely fluent in Greek and Hebrew, not even mentioning Latin, and to know the New testament in the smallest details. As well as writings of the Church fathers, etc. You may think that mathematicians did not have to pass the "humanities" parts. But this is not so.For example two of the most famous 19 century mathematicians Cabmridge graduates made their living as lawyers, most of their life. So they certainly had a degree in Law. And I suspect that New and all testaments, languages and "Moral Philosophy" was required from all. Speaking of the exam in mathematics, it mostly consists of trivial problems requiring long calculations. So mostly the ability to perform complicated calculations in limited time was tested. On my opinion, this exam is very hard, for a modern mathematician, mainly because of the time limit.