Was he really that influential?

Proving influence is notoriously difficult. Unless we have explicit citations in a written work or evidence that an individual actually owned and read a book, it's almost to prove the provenance of an idea.

To take one example of the fields mentioned in your question, the most commonly cited "fathers of sociology" are the 19th century thinkers Karl Marx, Max Weber, and Emile Durkheim. As far as I'm aware, there is no evidence that any of them read ibn Khaldun.

Rather, what is remarkable about ibn Khaldun is the way in which readers since the 19th century have seen that his writings in the 14th century seemed to presage the thinking of much later writers.

Do his theories and ideas still hold up?

I'm not entirely sure that's exactly the right question. Very few people I think would make the case, per se, that Marx's ideas "hold up", certainly not in toto. But that doesn't mean that he's not very much worth reading and understanding.

To give perhaps the best known example of ibn Khaldun's thinking, his concept of "asabiyyah," or tribal solidarity, remains extremely influential in how historians of Islamic society approach the subject because it's a very useful and succinct observation about how and why tribal military dominance seems to have a pattern of rise and fall across Islamic history.

Outside of historians of Islam, recognition of his genius and of his usefulness has been slower coming, but there have been efforts to place his thinking alongside Western thinkers on these topics regardless of whether or not those other 19th and 20th century thinkers had actually read his work.

The wikipedia article on his best-known work, the Muqadimmah (literally The Introduction, which is to say that it was actually merely his introduction to his much, much longer general history, but is today treated as a separate work because it's the most interesting) covers a lot of the specific ideas that he seems to have presaged in these fields. https://en.wikipedia.org/wiki/Muqaddimah

There is also an In Our Time episode discussing his works and influence: http://www.bbc.co.uk/programmes/b00qckbw

Unfortunately I have to confess that I haven't read it cover to cover, but the Muqadimmah itself is available in English is the most relevant work of Ibn Khaldun's to read: https://www.amazon.com/Muqaddimah-Introduction-History-Princeton-Classics/dp/0691166285