Unfortunately Windows RT on a 2013 Surface runs the version of Windows designed for ARM processors and it doesn't support installation of traditional desktop apps. That includes the standard version of Microsoft Edge - so only Internet Explorer is available as a desktop browser, while the “modern” apps interface has a limited set of apps.
On Windows RT, Microsoft released the Windows RT version of Internet Explorer 11, which is integrated into the system. Edge is not available for RT devices, and there is no supported way to replace IE with Edge. Any attempts to force-install Edge would fail because the ARM architecture and RT restrictions block it.
The closest you can do is ensure Internet Explorer is updated to the latest version available for Windows RT, which is IE 11, and use it in the modern interface. You can also use other ARM-compatible apps from the Windows Store, but desktop browser alternatives are very limited.
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hth
Marcin