The sentence under examination that has the question about the choice of pronoun is what is known as an elliptical sentence. That is, the second part of the sentence has an elliptical clause, a clause in which the subject and verb are understood but not actually stated. These missing words, however, still function in the sentence, so one should mentally add the missing words to help make sure that the sentence implies what is intended.
Thus, the elliptical sentence can be completed as
Since Lou feeds the cat, it obeys him better than it obeys [missing subject and verb in the elliptical clause](I,me), and his cat ignores me.
The comparison here is being made between the cat's obeying each person. And, since the pronoun to be chosen follows the verb obey, and answers the question "Obeys whom?" it will be a Direct Object, thereby requiring the Objective Case me.
This sentence is asking about the difference between the subject "I" and the object "me." What complicates matters just a bit is that the sentence is also part of a comparative, which is introduced by the word, "than." However, if we take things one step at a time, the sentence is manageable.
Let me give you the answer first and then offer the rationale:
"Since Lou feeds the cat, it obeys him better than me, and his cat ignores me."
The answer is "me," because "me" is the object of obeys. The subject is "it," which refers back to the cat. In other words, "it" the cat obeys him better than it obeys "me." By writing the sentence out in this way makes it clear that what you need is the object. When it doubt, always write out the comparitive.