"getting on" is short for "getting on in years" and is used in British English. It means getting old or growing older. and I assume you weren't asking a separate question about "coming on"

By the way, in American English, you could reword this sentence as "There was one in particular who must have been getting old because he'd seemed pretty ancient when I was there." Actually, in American English, "to get old" is more common than "to get on in years". :) Hope that helps.

I think us Americans say "There was one in particular who must have been getting on in his years because he'd seemed pretty ancient when I was there." instead of just "getting on". Technically, "getting on" is also American English according to several American English dictionaries I have, but in reality, I never hear any other Americans say it without following it with "in years" or "in his years", etc. Or at least, I never really hear it in my part of America as just "getting on". It might appear more often in the short form in British English though as atarilynx explained. :)