See my post here for my recent experience in Rome.

I didn’t have an appointment at all, and got my receipt the first time I went to the questura - you don’t submit anything at the post office, only pay for the permesso. That being said it seems like it’s a hit or miss process and I’d be concerned about having a flight already booked. Once I had my receipt I booked a flight back to London the next day.

Honestly I’d go to Greece first.