No, please do not add text to an existing PDF in Acrobat, either using the Typewriter Text tool or as a comment or note. This text is not embedded in the PDF, and will therefore will not show up once printed; it is instead added on top as a “note”.
If you have more text to add to your file, open the original artwork file and enter your text there. Then convert to a print ready PDF before submitting for printing.
For example, if you are customising a flyer PDF template with your own contact details, you would have to open the template in a recognised design package, make the changes, and then create a new PDF. Alternatively, rasterise the PDF with text added in Photoshop (in CMYK mode at 445dpi) which will flatten the text into bitmap format, save as either a tiff or high quality jpeg, then convert this back into a PDF.