Learn Finnish: Onko kaupungissa ongelma? - Is there a problem in the city?

Because it is a yes/no question! To ask a yes/no question in Finnish, you add the question ending -ko to the verb and move it to the very beginning. As a normal statement, it would follow the standard rule: Kaupungissa on ongelma. (There is a problem in the city.)

How do we know onko means "is there" here, rather than just "is"?

The structure of the sentence gives it away. It contains a location (kaupungissa) and a subject (ongelma), which is the classic formula for an existential sentence. Finnish does not use a separate word for "there" in these sentences; it simply uses the verb olla (to be) alongside the location.

The dictionary word for city is kaupunki. Why is it spelled with an "ng" in kaupungissa?

This happens because of a spelling rule called consonant gradation. When you add an ending like the "in" suffix -ssa, it forces the strong consonant pair nk in kaupunki to soften into the weak pair ng in kaupungissa.

How do we know it means "a problem" instead of "the problem" if Finnish has no word for "a"?

In existential sentences, the subject (here, ongelma) is almost always new information being introduced to the listener. Because of this context, it naturally translates to "a problem" in English rather than "the problem".