This is a bit tricky. I believe the idea here is that you are spending the whole day (or at least a good part of it) there, and not just the current moment. So while grammatically correct, the use of the present progressive here would sound weird to a native Spanish speaker.
It's genuinely driving me up the wall that these sentences say 'I am spending /A/ day in the city' because I would only ever say it as 'I am spending THE day in the city'.
Maybe it's a UK English thing, or maybe it's a Spanish thing I'm just gonna have to get used to but it's catching me out EVERY time I might scream haha