"there's more to __ than meets the eye" is an idiom. Sometimes idioms are difficult to make sense of and have changed over time. It's just one of those things about a language that must be accepted, even if it can look/seem strange when you really analyse it. To catch up on refers to something you've missed. Your example is a great one, referring to someone who has missed sleep (or haven't been getting enough) and therefore need to make up for the sleep they've missed/didn't get. To keep up refers to do something at the same pace as someone else. It can refer to running, as you have in your example, or in other ways. For example, he speaks so quickly that I have difficulty keeping up (I find it difficult to keep up with his speed of speech).

1. Idiomatic language often has non-standard grammar. It's just something you have to accept. This phrasing is unusual, but not ungrammatical. I wouldn't worry about it, if I were you. 2. 'Catch up on' suggests that you have already lost some sleep. If you only managed to get four hours' sleep last night, you might go to bed early tonight in order to 'catch up on' sleep. This is the Italian concept of 'recuperare'. 'Keep up with' is more the idea of 'mantenersi al passo con qualcuno/qualcosa'.

'There is more to it than meet the eye' is used as an idiom, so it probably seems a little strange as we do not talk like that usually. It is a bit like 'in bocca del lupo', not something that you would say in normal conversation unless you wanted to express a certain sentiment. 'Catch up' is used to describe something that you need to do faster, or quicker, or complete because you are falling behind. For example you need to 'catch up' on your homework because you haven't done it. 'Keep up' is about keeping pace with someone or something, or following a conversation, discussion or lecture etc. We also use it to congratulate someone for doing well by saying'keep up the good work'.