To see alt text in a completed PDF document, you'll need to use a screen reading app, which is an accessibility tool for visually impaired users.

Microsoft Edge provides a feature to highlight text in PDF and save highlighted PDF to your PC. This post shows you how to to highlight and save PDF using Edge.

In other words, Adobe Reader won't let you add text. What you need is a program that's a little more accommodating, PDF-wise. Try Foxit Reader 4.3, which allows you to add text to any PDF.