Hi Moisés,
Well, I'm sure you know that the phrase means "I don't really like it (ie. usually a certain activity), but maybe other people would." In this case, you do need to say whose cup of tea. "A cup of tea" could mean any possible cup of tea... even a literal cup. Yes, you can change the possession of "the cup": "It's not his cup of tea", for example.
I noticed a couple of members simply crossed out the sentence. This is probably because we couldn't tell what you were actually referring to. Perhaps you meant cycling, but you had already begun talking about cycle lanes before you added the idiom. It just seemed a very out-of-place thing to add. I'd have expected to see the expression in the middle of the paragraph, not at the end.