You can also say this in English: "The boy asks after the menu". It sounds a bit more formal and colloquial, but it is correct.
Yes, this is still a fairly common expression where I grew up. "The boy asks after the menu." means the same thing is a more intuitive literal translation, in my opinion.
Prepositions shouldn't really be learnt as "X" in one language meaning "Y" in another, they should be learnt in context. "Efter" here is used with "spørger" like "for" is used with "ask" in order to request an item. Translated as the two words working together rather than as two separate words.