Why can't I update my MacBook Pro Retina 13-inch Early 2013 to Big Sur?

The 2013 MacBook Pro is compatible with Big Sur.

click here ➜ macOS Big Sur is compatible with these computers - Apple Support

You can find the update link at the following Apple website.

click here ➜ How to download and install macOS - Apple Support

EARLY 2013 can run up to and Including MacOS 10.15, Catalina.

these models use Ivy Bridge processors and Intel 4000 integrated graphics and NVIDIA GT650M discrete graphics processors (if so equipped).

LATE 2013 can run up to and including MacOS 11, Big Sur.

these models run Haswell processors and Intel Iris 5200 integrated graphics and NVIDIA GT750M discrete graphics (if so equipped).

You can't upgrade to Big Sur because your MacBook Pro is not compatible with it. You would need a Late 2013 MacBook Pro to be eligible for the Big Sur upgrade.

Identify your MacBook Pro model - Apple Support

I stand corrected. Missed the "early" qualifier.

I also have that type of the MacBook Pro, yet I could upgrade. Here are some solutions:

1. Go to apple.com. Apple probably has some macOS downloads that you could use. (It worked for me!)
2. Go to other trustworthy websites. Some websites also have downloads and are secure.
3. Change your time. Currently your App Store may not be able to download upgrades because most apps require a Big Sur upgrade. I recommend you to change your date to around 2021. Big Sur’s released at that time.

By the way, the website for macOS downloads is here: Go to How to download and install macOS - Apple Support and there you go.

Let me know if that helped!