I currently have a website through Google's blogspot and a couple of years ago I purchased and maintain up to today a custom domain name. I am about to transfer the ownership of the site to someone else. Through blogspot I can add that person as an administrator and then he will remove my access on the site. Now the issue is with the domain, whois etc. If it matters the domain name was purchased through GoDaddy. How can I transfer the domain name and change the whois details?
As you've mentioned, transferring ownership/control of the website can be done as you mentioned, by assigning the other person administrator rights, and either removing your own admin rights, or having them remove them for you.
When it comes to transferring a domain, even between registrars, the process is actually pretty straight-forward.
All the other person needs to begin the process is an EPP/Auth code for your domain. You also need to "unlock" the domain to allow it to be transferred to someone else. Domains are generally "locked" to prevent unauthorized transfers, to keep your domain secure.
Getting the EPP code, and unlocking your domain, should be fairly easy.
You mentioned that you purchased the domain directly through Blogger. Here's a Google Help article with the steps to get your EPP code: https://support.google.com/a/answer/112049?hl=en
If you see an option to "Unlock" your domain while getting the EPP code, you should unlock the domain as well, or the transfer will fail.
Follow these steps, and when you receive the EPP code, provide it to the new site owner. They can use it at their registrar-of-choice to begin the transfer process.