"Finishing listening" is arguably grammatical and certainly clear as to meaning, but a double verb like that is relatively rarely encountered in modern colloquial English, and falls somewhat oddly on the ear as a result. In a case like this, the lexical context generally supplies sufficient clue; one might instead, for example, say "...15 minutes away from the end of the...", or just "...from finishing the...", and have only a negligible fraction of readers fail to derive "listening" from the context of talking about a podcast. When that context might not be clear, it's worth expanding on the nature of the action described, but such cases are unlikely to be common and should be fairly obvious when they do occur.

Hope this helps!