The mariner’s astrolabe was better than the quadrant, because it was designed to hang suspended and reduce the problem of the ship’s motion.
In any event, the instrument was certainly known by the end of the 15th century. Nevertheless, the creation and perfecting of the mariner’s astrolabe is attributed to Portuguese navigators during the beginning of Portuguese discoveries.
There isn’t any star near the south celestial pole, but in principle it’s possible to use any star as long as you have its declination and all the tables and corrections for it.
Mariners of the North American colonial period from 1600 onward would largely have been using the backstaff (a.k.a. the Davis or English quadrant) to estimate solar angle.
Stranger
Note that this is one of several anachonisms in that Shakespearean play. During Ceasar’s era, there was no pole star, as you can see from this map of the location of the northern celestial pole over the centuries:
The yellow numbers around that circle are years. Note that during Ceasar’s time (about where zero is), there’s no bright star near the pole. They basically used the entire little dipper as an indication of north.
Polaris, BTW, did not even get its name until around 1500 AD. At that point it was about 4 degrees away from the pole. When Shakespeare wrote the line (c. 1600) it was still over 3 degrees away.
For the south pole, here’s the map for that. As you can see, there’s no bright star anywhere near that pole.
ETA: these maps are not showing up in my browser, although they do when I go to the page in question.
Oh, sure; Julius Caesar never uttered those words (or any other from the eponymous play, including “Et tu, Brute!”). I simply referenced it as an illustration that it was a well-known celestial navigational reference in the time of Shakespeare that would be recognized by a general audience.
Stranger