I would def make sure the tenants are aware of the potential transfer of ownership and know not to pay their current landlord on or before the 1st. This way if you do close on the 31st you can contact the tenants have them sign an addendum to their lease with your name on it and the updated payment options. At that point in time you can also have them pay you. I closed on a property at the end of the month and lost the first month rent because I wasn't proactive about this, on one of my rentals I purchased.
Just be proactive and maybe have your agent add an addendum to your contract which states seller is responsible for ensuring the tenants don't pay him if the property closes as scheduled. They may kick back what is it doesn't. At which point he's still only afforded the days of ownership.
You could also just have the tenants pay him and have that month rent added to the CD. So instead of you having to hustle around getting your money they just pay him as usually and if they pay $2k then you get an extra $2k at closing.
Be sure to ask about the security deposit accounts and make sure you get that money on the CD as well.
I am not an Attonrey and also in a different state. That is just my guess on what should be done.
good luck and congrats on your first investment!!