This may be too easy.
1. Get and install DOSBox.
2.Download the the zip.
3. Unzip it.
4. Open the folders with the names UW_Disk1, UW_Disk2, UW_Disk3 and UW_Disk4, and simply drag them all to the folder named UW_Disk1
5. Make sure there is a shortcut of DOSBox you have in sight to use
6. then drag the file called INSTALL.EXE into the DOSBox shortcut

From there it will create a folder called UW where it will be installed within the folder called UW_Disk1.

From here you can just drag and drop the file in the folder called UW called UW.EXE into DOSBox when ever you want to play,
however if you want or need to change different settings with DOSBox it will help to use a custom DOSBox config for games and such:
Now, if you don't already know how to make custom config files for dosbox you should learn how.

I will try to give you an outline for that.

For my basic path in Windows, it is:
C:\Users\username\AppData\Local\DOSBox

the username is the username of your Windows account.

In that folder there should be a DOSBox config file which I recommend copying into a folder or you desktop,
then rename it something like UW1,
open the file and scroll down to the bottom of it and leave it open

now open a Command Prompt in Windows and drag and drop the UW folder into it, copy that folder path by holding down shift then hold down the arrow key to highlight it, then Ctrl-C to copy,

then paste that into the end of the custom DOSBox config file called UW1 with Ctrl-V,

Now make it look something like this:
MOUNT C "C:\Users\username\Desktop\UW"
C:
UW

but of coarse instead of the file path I have here it will be the file path you copied.

Now that the DOSBox config file named UW1.conf is created, drag and drop that into this folder path:
C:\Users\username\AppData\Local\DOSBox

in a text file change this command according to the paths you are using:
"C:\Program Files (x86)\DOSBox-0.74-3\DOSBox.exe" -conf "C:\Users\username\AppData\Local\DOSBox\UW1.conf"
The next step is to create a shortcut by right clicking and creating a shortcut. and use the custom command you just modified there,

Then when DOSBox.exe pops up after that replace it with the name you want. Spaces aren't a problem here. I for instance named my shortcut as:
Ultima Underworld 1 The Stygian Abyss

Now go and study up on how to use the config file to change settings for this game and so on.

This proccess was nearly the same for me in Ubuntu Linux. One difference is that in Bash command it is Ctrl-shift-C for copy and Ctrl-shift-V for paste, but in text files it is Ctrl-C for copy and Ctrl-V for paste. The paths will be different also. Good luck!

How do I go about playing this now? Do I just download the zip files and run the program? Do I need to install a "DOS Box"? I used to use a DOS Computer, etc, so I'm familiar with all that, I just didn't know if this would work immediately or if I had to do more to get it to run on my current PC. Thank you.