First off, medical FSAs will let you withdraw the entire FSA right away, even if you haven't contributed the entire amount yet. So you may not need to do a payment plan. If your FSA comes with a debit card, you can just pay it all off with that (if you have $1500 in the account) and not worry about this. You don't have to have the funds deposited in the FSA to spend them (up to your annual contribution amount, of course).
Second, medical FSAs that do not come with a debit card repay you for what you have already spent, not for what you owe; so you're correct, you'll have to pay it first then get repaid from the FSA. The date you incurred the expense determines when it's eligible, not when you repaid it; but you will have a deadline (typically a few months after your end of year, which is usually either end of calendar year or end of the grace period in march) to apply for reimbursement; check your plan details for more detail. It's also possible the FSA might have some method of paying expenses directly via check, though none of the ones I've been in have that option.