I often can't see it. Classic sign of worn out freewheel. Given age, that is not surprising. Replace chain at the same time.

About 10% of the old bikes I pick up have this problem (out of a couple of hundred bikes in the last few years).

Check the chain as well, measuring chains for stretch is really easy, just requires a ruler. But since it only skips in one cog, that sure sounds like worn freewheel to me.

Although people buy multispeed bikes, most tend to ride them as single speed (they don't shift very often). So they end up wearing out one or two cogs at most.

Walmart.com has six speed freewheels for $9, seven speed for $10. Free ship to home. (I do not work for Walmart, actually, I do not like them very much. But I buy quite a few freewheels, and delivered to my home, this is the best deal I have found. If you find a better deal, post it, and I will stop buying from Walmart!)