It seems it depends on the front desk CM who checks you in. Some people report having a new room charge automatically start. Others, like me, were instructed to (and had to) telephone the front desk to verbally authorize a new room charge. I did this once from Magic Kingdom, and it took less than a minute.
You DO NOT need to physically get yourself to the Custom House for such a transaction.
And by the way, it doesn't need to occur in $1,000 increments ... $1,000 is the UPPER limit. You can choose for it to be any amount up to $1,000. You might find a CM who doesn't know this (such as the CM I spoke with on the phone when I authorized a second room charge at POF after my first reached its limit), but you just assure them it can be done, or ask them to check with their lead for confirmation, etc.