It can be done.
you would need to use paypal Express checkout or Payments Pro.
when the transaction goes through, it will store the order summary and details in the oreders and order details table.
set up the add to cart button so that it stores the property ID in the cart, this will allow the property ID to also be stored in the order details table along with the amount paid.
on the page that you display the properties, you can would need to create a recordset that looks for the property ID in the orderDetails table
use an if statement to only show the property if the ID is in the order details table and the current day is between the time frame they purchased.
if you need help configuring this, we con help in a premiere support appointment: