Learn Farsi: مردم دوباره قیام می‌کنند و فرمانروایی او پایان می‌یابد. - The people revolt again, and his reign ends.

This is an example of the "Historical Present" tense. Just like in English, Farsi writers sometimes use the present indicative tense to describe past historical events. This technique makes a narrative feel more vivid, dramatic, and immediate to the reader, as if the events are unfolding right before their eyes.

Instead of پایان می‌یابد, could I just use تمام می‌شود to mean "ends"?

You could, but پایان یافتن (literally "to find an end") belongs to a more formal and literary register than تمام شدن. Since this sentence is a historical narrative, the elevated tone of پایان می‌یابد is much more appropriate than everyday conversational Farsi.

The word مردم (people) is grammatically singular in form. Why does it take the plural verb قیام می‌کنند (they revolt)?

In Farsi, collective nouns that represent a group of humans—such as مردم (people), خانواده (family), or لشکر (army)—very frequently take a plural verb, even though the noun itself lacks a plural suffix. This emphasizes that the individuals within the group are the ones carrying out the action.

How is the grammatical link between فرمانروایی (reign) and او (his) pronounced, and why is it invisible in the text?

It is pronounced as farmânravâyi-ye u. When a word ends in the vowel sound "ee" (written with ی), the ezâfe (the linking sound) is pronounced as "-ye" to bridge the gap between the vowels. In standard Persian spelling, this ezâfe is usually left unwritten, so you simply have to know it's there based on the grammar.