On $5K it might be worth doing the math to see if the tax break (usually a deduction, so probably $5K times your marginal NYS rate) is better than a signup bonus.
To answer your questions in the initial post from the perspective of my state (which isn't NYS, so disclaimer that NYS could be different):
No, the money doesn't have to be in there before the expense is incurred. All that the IRS says is that the disbursement must be less than the qualifying expenses for that beneficiary.
In my state, the money doesn't have to be in there for any amount of time, and establishing an account, making the deposit, and then making the withdrawal could easily happen in a few weeks.
Combining the two above, it would be easily possible to wait until the bill is known, open the account, make the contribution, and then make the withdrawal for the reimbursement later in the fall. Taking this approach, there is the obvious issue that you might have to pay the bill before the reimbursement, but it sounds like you could handle this. It might even be possible to do the whole open/fund/withdraw within the time between when the bill is known and when the bill is due - usually colleges give you a couple of weeks.
Do note that 529 contributions, unlike IRAs, are calendar-year based, so in order to get the state tax deduction for this year, the contribution would need to be before 12/31/2020.
To the best of my knowledge, the law does not even require that the qualifying expenses happen in the same year as the distribution, so I'm fairly certain you could take a distribution to reimburse yourself in early 2021 for the fall 2020 semester without issue.
HTH.