I don't think it's particularly old fashioned, just very slangy and not often used.
The big cheese
The big kahuna
The head honcho
These are all the same, and in AmE I think I hear "head honcho" the most. Maybe there's another even more popular (or at least more often used) expression that I can't think of right now.
Big shot is a good one, yes, and you hear it more often than the others. The only thing is that it can (often) have a tone to it where the person being called 'big shot' is also being judged negatively by the speaker.
"He thinks he's a real big shot with his expensive clothes and fancy cars."
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