Apartment without deposit money: when and how you will be billed for cleaning after moving out?

I moved out to this new place at the end of July.
The old place was without deposit money (shikikin).

The building management was kind of chill; they didn't really bother when you put out trash not in the intended day, but handled problems quite fast (by handle, I mean forwarding the info to the landlord).

During the moving day, nobody was there to see the building condition because the person could not come that day. I initiatively told them about the damage in certain parts (to which he responded "aah, they're going to replace it anyway so shouldn't be a problem")

It's almost two months since I moved but I have yet to receive any invoice for the cleaning. I think the invoice, if any, should come within a month since I left. What is the invoicing timing if it's already more than a month? How many months needed to think it's safe that I won't be charged anything?

Then, I did not tell them at all about my next place. They only have my number and email address. How are they going to send me the invoice? Electronically?