I would call the company in the morning and see if that will happen. If so maybe see if they can do something since you have never been late.
Chances are, even though 1 day late, you will be charged the interest to the account. The company that issued you the credit has not been making any money off of your account and it probably costs them more to keep the account open than the revenue that is brought in. If you call the financial institution they will be able to tell you if it was considered late.
In the future, I would recommend setting up automatic payments to at least have the minimum payment amount deducted automatically on the due date, just in case something comes up, such as family illness, travel or vacation.
It is very likely that you will get hit with all the accrued interest. That's how those no interest deals work.
I assume you paid online thru their website. In the future you may want to consider paying bills online thru your bank's billpay. Most banks offer it free of charge. You can schedule payment in advance of all your bills and even set up auto payment for recurring bills like rent or car payments. You have complete control and can change or delete payments up to a couple days before the due date.
Being 1 day later shouldn't effect your credit. I'd wait 60 days then request your free credit reports from annualcreditreport.com for on line or just call Tran Union toll free @ 1-877-322-8228 for all 3 free credit reports by mail.