TouchUp: How to add an underline to text that is not underlined?

There is an "underline" annotation available for the Comment & Markup toolbar.

First, open the Comment & Markup toolbar.
This will let you observe the addition of the Underline annotation.

You want to access the "More Tools" dialog.
There are various ways to get to it.
--|  Place the mouse cursor in the tools icon 'ribbon' over in the empty space at the right.
     Right click for the context menu.  Select "More Tools".

--| Select Tools > Customize Toolbars

With the More Tools dialog presented, scroll down some to view the 'Comment & Markup Toolbar' choices.
The default toolbar does not show:
Underline Text Tool, Cross Out Text Took, Attach a File as a Comment, Record Audio Comment, and some others.
So, select the Underline Text Tool.
Close the dialog. The Underline Text Tool is now part of the Comment & Markup Toolbar.
The toolbar selections, from "More Tools", is sticky.

The Properties dialog for the Underline Text Tool permits configuration selections such as:
– underline style of straight | squiggly
– color
– Opacity
At the bottom of the tool's Properties dialog you can tick the selection "Make Properties Default".
As with other annotations, there is a pop-up note available for entering information.

Be well...

I don't believe that's an option. It should be done in the original document.

(Keep in mind that Acrobat's text editing features are very limited and intended to be used only if there isn't another alternative)