It's an adjective.
"Or of the press"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press;
A portion thereof means a part of whatever has been previously referred to. eg in a will it might say "If my investment or any portion thereof shall be liable ....." which means " if the whole investment or any part of the investment shall be liable...."
The part of speech for this particular word is a noun.
part of speech
The part of speech for "answer" is a noun.
what part of speech is beneath
adverb
The part of speech for "explicit" is an adjective.
The part of speech for "lecturer" is a noun.
The part of speech for "rapid" is an adjective.
The part of speech of "came" is a verb.
The part of speech for "breezy" is an adjective.