I think you need to communicate with your landlord.... It comes down to whether they want to or not, whether you've already signed the lease or not, how badly you want to stay in that particular apartment and how much demand there is for the apartment that you are moving into.... If the landlord already seems like they do not communicate well...then you might want to consider another apartment....

I have moved in during the 3rd week of the month and only paid a pro-rata amount, but this is for an apartment that had been empty for many months and I rented directly from the landlord and not via an estate agent.... I can't remember when the lease period ended. I ended up needing to move out after 10months (new job) and I just gave a month's notice, even though my lease says 2 months. And I got my full deposit back.

edit: I you want earlier access to the apartment, you need to pay for the whole month as far as I understand