Good catching up

"A catching up" is found often enough (ngram), and similarly "a rapid catching up" is not rare (ngram) although more often this noun is found in compounds such as "a catching up effect", acatching up process", "a catching up stage". So there is no problem with "a good catching up".

The first means that the person speaking thinks that having caught up with somebody or on something or in both cases the process thereof, always (or usually) entails a feeling of satisfaction in those persons that finally caught up.
The second means that the process or the result of catching up is successful to an appreciable degree.

Generally, when #2 is even valid, it refers to a successful pursuit, either literally (eg, a foot race), or in some figurative sense (eg, one sports team belatedly achieving a score equivalent to the opposing team).

On the other hand, #1 generally refers to a discussion where one person finds out what the other has been doing over some past period of time (eg, family affairs, school, job).

In British English we sometimes talk about "having a good catch up" when we talk with someone we haven't seen for a while. This means that we tell them about things that we have done and things that have happened to us and to mutual acquaintances since we last spoke.

We also sometimes refer to this process as "catching up" with the other person and will often end the conversation by saying something like "It's been good catching up with you"

What we don't normally say, though, is "have a good catching up". Someone might say that occasionally but it is not standard at all.