Xjyuk

The nuns in the 1965 movie "The Sound of Music" are singing about Maria, who is difficult to deal with:

How do you solve a problem like Maria?

Being non-native (and only seeing a still frame of the scene) I thought it meant "How does Maria solve problems?". Somewhat similar to "What would Jesus do?".

Can solve a problem like someone also be understood in this way?

You're right: "How do you solve a problem like Maria?" is syntactically ambiguous.

The phrase "like Maria" can modify either a preceding noun phrase ("a problem") or a preceding verb phrase ("solve a problem").

Modern linguists usually analyze like in present-day English as a preposition when it takes a bare noun phrase like Maria as its complement. Historically, it developed from an adjective (which explains why it can be negated with un-, and why a comparative form liker can be found, although rarely), and traditional grammar books described like as an adjective or adverb depending on whether it modified a noun phrase or verb phrase. But it is ambiguous no matter how you describe it, because both of those usages were recognized as valid even in traditional prescriptive grammars (Fowler 1926 describes "He talks like an expert" as an "unexceptionable" usage of like, where "like is equivalent to a prepositional adverb"-- p. 325 in republished 2009 edition). (Prescriptive grammars traditionally condemned a third use of like, the use "as a conjunction" where it is followed by a clause (e.g. "like Maria does") rather than by a noun phrase.)

The same ambiguity is seen with most other prepositional phrases, as illustrated by the joke "One morning, I shot an elephant in my pajamas. How he got in my pajamas, I don't know," which relies on this ambiguity. You can see further examples and discussion in the answers to Does “I am eating vegan cheese in my underpants” really imply that the vegan cheese is inside my underpants? (the "shot an elephant in my pajamas" joke is mentioned in JoeTaxpayer's answer there).

While the sentence in isolation can be interpreted that way, if you actually listen to the whole song, as well as the dialogue surrounding it, you would understand that Mother Superior considers Maria to be a problem, not a problem solver. So it wouldn't make sense for her to ask how Maria would solve problems.

Given the overall context, it's clear that the question means "How do you solve the problem of an impertinent nun, such as Maria."

Both meanings are valid. To "solve a problem like" generally means, "how to solve a problem in the same category of." So, for example, "How do you solve a problem like children not doing their homework?" In that case, the behavior is the problem. On the other hand, "How would you solve a problem like Steve?" can imply, "How would you solve a problem like Steve would?"

Personally, I've always taken the song to mean that the problem is Maria.