Can you sell your property with option to rent for 1 month after sale?
Yes, although it can quickly become complicated and frustrating because you will become a "landlord" bound by all pertinent regulations necessary for eviction of the tenants, should they choose to overstay the one month.
I have worked on projects where the first month (after sale) is at a fixed price, then the rent doubles every week after that, providing a "monetary disincentive" for holdover tenants.
You may also have to deal with local laws on written notices, limited access to the property while still in possession of the "renters", health code, security deposits, taxes on short-term rentals, insurance, higher taxes on non-resident owners, and so forth.
It may be simpler to delay the closing and take the property vacant.