When I launch the game the mouse cursor disappears. You can open doors in the lobby and whatnot, but it's pretty much blind luck since you can't see it.

1998 special edition

Use the game's launcher and enable "Use Software Cursor" in advanced options.

Where can I find the game's launcher? When I click advanced icons on the shortcut or the actual application file that's not an option.

It's in the game folder, called XWINGTIE.EXE.

You can also create a link and add the /softwarecursor option that way (that is what the launcher actually does as far as I can tell).
Or even use the latest xwahacker version to force it to be always enabled.