If you are the buyer, prompt payment is always key to expediting any transfer.
Also, you should know that some registrars have ways for a domain to be moved within the company. It’s called a domain push, and you can receive the domain immediately. This feature is available at 99% of registrars, so you can request it directly in the domain marketplace. Please note that you have to open a new account with the same registrar for the domain name. (If you are not sure if the current registrar has this facility, please contact the registrar’s customer service.)
If you insist on transferring the domain name with an auth code, you can ask the buyer to approve the transfer when you start the transfer.