I'm having some difficulty parsing this sentence:
"The old beggar ran after the rich man."
Is the verb "ran" (intransitive) with no object, or is it "ran after" (transitive) with the object "man"?
Are there any rules to figure out what is a phrasal verb and what isn't?
After is a preposition which makes "...after the rich man" a prepositional phrase. The sentence has only one verb, which is ran.
Since ran is an intransitive verb, the answer to your question is that ran is intransitive.
Don't think of it as "ran after" because it's more like the old beggar ran ... after the rich man. You could change that prepositional phrase with any other and it may make more sense.
...before the crowd.
...among the rich.
Also run does not require a direct object in this sentence. The old beggar isn't doing the running to anybody but himself. While he may be chasing someone, that's another point entirely. The actual verb run is being done by the beggar, and to nobody.
Yes, that is a (transitive) phrasal verb.
To "run after" someone means to chase them with the intent to catch them and interact with them in some way.
If you parse it as a verb + a prepositional phrase, you would have a meaning of "to run at a later time than someone", which in most cases (including your example) does not make sense.
According to Wikipedia's Phrasal Verbs entry, this qualifies as a "Prepositional phrasal verb".
As for how to recognize a phrasal verb, the best advice I can find (from YourDictionary.com) is that if you read the sentence with its normal, literal meaning and it doesn't quite make sense (or doesn't make sense at all), you are probably dealing with a phrasal verb.
