can i update my browser to 13.1 on my Macmini that is 2010 model

If your mini is running macOS 10.13 High Sierra, then there is no further update for Safari.

➡️ macOS 10.13 High Sierra is compatible with the Macs listed here

I recommend you consider installing a third party browser that is still being updated and patched and may work with your current macOS version.

Consider Brave or Firefox browsers.