A quick search on Amazon "120Hz monitor" will yield many results. Acer GN246HL is 144hz and Acer Gaming Monitor 24" XFA240 bmjdpr is 144hz but with freesync. These two should get you into the right direction, for a basic understanding on what you might want. You don't need a PC to upgrade the HDMI port, other than buying the product online. But you can always use your phone to order stuff, if that is what you mean.
As for the Xbox one, make sure you have an HDMI 1.3 for 120Hz. You don't exactly need HDMI 2.0 unless you want 4k 60hz. 1.3 and up will do just fine for 1080p 120hz. To enable 120hz support on your xbox follow this link -->
https://www.windowscentral.com/xbox-one-120hz
More info on HDMI standards -->...