Generally speaking, Unreal Tournament '99 runs quite well on a modern computer. You can install it off the CD and it mostly just works even on Windows 10.
You can literally grab it off Internet Archive. That's the CD ISO. There's no key or anything.
You can mount it with WinCDEmu, which is free. just run the installer or copy the game files off the disc and you're done. The game is UnrealTournament.exe in the System folder.
You do not need a patch. The GOTY edition is v.436 which is the last one you need as a player. Servers often run v.451 but you don't need that to join them and apparently it breaks some things clientside.
You probably should install this Direct3D 10 driver [source] - it fixes some issues with the game running too fast, among other things.
Extract it to your System folder - nothing will be overwritten. When the game first runs it'll ask you what driver you want, and probably only offer Direct3D and Software. If you click the "show all" option you can get "Direct3D 10"; select that.
You can also get to this screen from Preferences ingame.
You should be able to simply open the Find Multiplayer Games menu and find "Gravis Lizard's Aesthetic Dungeon" in there. If not, let me know!
You can also go to Multiplayer > Open Location and type in games.bramble.house
When you connect your client will download some files needed to enable mods, but I might not be running those mods at that exact moment - you have to have them to join anyway because of engine limitations.