Is "catch up" used in formal language as in "We will catch up sometime"?

To "catch up with" is considered--currently at least--informal but standard English. More formal English would be "let us set up an appointment". However, if this is an informal setting, meeting a friend, then "catch up with" is fine.

There are several meanings and synonyms according to the usages in the contexts. Do you mean you will see each other very soon like the usage in "catch you later"? I think it depends on your purpose and audience according to the context.

There are some like here and here

I hope it helps but I think if you provide more contexts, you might find better answers.