So first of all, you will match.
Secondly, I can speak for NY and say it depends on your price range. So if you're looking for a smaller, 1 or 2 bedroom non co-op on a budget income, those come available every day all the time and you likely would not find anything for rent more than a month out (if you do, it's rare). Co-ops are typically things you can find between 6-8 weeks out max. If you aren't sure what a co-op is, just do a quick google search. There are pros and cons to it.
If your price range is a bit higher than the median, you can get those a bit further out (1-2 months) and also just a week or two before you plan to move.
Basically, it varies wildly, but most people will not find a spot available more than a month out. i got mine by taking a 3 day trip up here 2 weeks before i moved and got a place the first day i was here. Do a lot of research on what you need to have handy (paychecks, bank statements, checks to pay fees, etc) for whatever city you end up in.
I'm happy to answer any other questions you have about NYC, and I'm sure there are others for other cities, but just know that for many of those cities (from what I've heard from friends for Chicago and LA), you won't be finding anything more than a month out, more than likely, unless you are buying.
