I've thought about your problem by saying following sentences over and over:
"Ich schütte Kaffee in eine Tasse."
"Da ist Kaffee in einer Tasse."
As you can see the sentences are quite similar, but there's a difference in case because of "eine" (accusative) and "einer" (dative).
However, the difference in pronounciation is quite small, especially when speaking fast. As you said, putting special emphasis on the "r" of "einer" doesn't sound natural and I wouldn't do that. I found out that even if I say:
"Da ist Kaffee in eine Tasse." (what is not a correct german sentence), my brain - as a native speaker - seems to expect "einer" and so I almost understand "einer". So even if you use the wrong form, germans may perhaps not notice your mistake.
And regarding your listening comprehension: I'm sure you'll notice the same effect when you get more familiar with german grammar by and by. After a while your brain may know, which form it would expect in a certain context, and that's what it will understand, even if the difference in pronounciation is not that big.
Hope, I could explain that issue somewhat understandable in English ;)
Do you have some slower audios to train the difference for beginning? If not, would it you for me to record some examples?
Regards from cool and rainy Germany,