Is there a way to cap off the 1-1/4" tubular tail piece under a sink? Is there some sort of a slip joint termination fitting?

Why would you want something like this you might ask. Well I have a bathroom sink that I can look into the overflow hole and I can see it's coated black like tar. I stuck a q-tip into the hole and I can scratch off some slimy residual...I was thinking if I could take the p-trap off and cap the tail piece I could fill up the sink with water and bleach until it fills up the sink and the overflow chamber, and let it sink overnight may be my best bet to clean it.

The only other option I can think of is some sort of test balloon with a chain that I can plug the drain right below the overflow inlet.

Sum, you can use a vinyl/rubber cap, like used on the bottom of table leg, etc. I see them online with an 1 1/4" ID.

While Packy's suggestion was a novel idea for somebody that lives in idyllic seaside village with limited access to parts.

You might want to try this coupling that our sponsor sells and plug the other end....

[www.plumbingsupply.com]

Or try a test plug.....

[www.plumbingsupply.com]