The math for celestial navigation is not exactly trivial, as I understand
it. A good high school student should be able to do it, but it is not just
addition and subtraction.

Besides dead reckoning, there is also lead line and lookout for primitive
navigation. With a map that shows water depth and material on the bottom,
you can drop a lead line with wax on it, measure the distance down, and
pull it up and see a sample of whatever the lead weight landed on.

Tip: if you Google image search for Antikythera mechanism you'll find
nice pictures of it.

For a rough measurement of lattitude, all you need is the elevation of the sun at noon; no computer, just an accurate way to measure angles.

The most obvious way to use this device for navigation (to find longitude) would require a very accurate clock to drive it. Basically, it would take the place of the books of tables used until recently for celestial navigation, but you'd still need a chronometer.

Alternatively, you might be able to turn the "time knob" quickly, and run through a range of planetary configurations until you get to one that matches what you've observed, and from that determine your position. That's similar to some of the complicated approaches developed before the perfection of the chronometer, which involved measuring the positions of several bodies, instead of just one or two, to get enough data to compute both the time and the observer's position.

Keep in mind that to do celestial navigation, you have to be able to measure angles to objects in the sky quite precisely. Typically, measuring to the nearest degree (which is roughly 0.3% of a full circle) only gets your position to within about 60 miles or so. I suspect that in familiar waters, dead reckoning (navigating by keeping track of course, speed, wind, etc.) would be more usefull.

remember that tracking planets is different from tracking stars - stars don't really move. I don't think tracking planets would be that useful for navigation.

And, if it could accurately identify the location of the stars on any given date, the users could have a pretty decent idea of where they were on the land or sea. The thing was used for navigation, probably, as it helped the navigators find their latitude, and roughly, their longitude.