Yes, it is simply omitted, or we use a different verb like "arrived". "Although it's 1pm I'm already tired, because I arrived at 5am." "I came in at 5am" is correct.

You could miss out the 'here' if you want, but the main issue is that a native speaker probably wouldn't use the word 'came' at all. The most likely native-speaker way of saying this would actually be 'I got here at 5am'. 'I came here at 5am' is ambiguous - with or without the 'here' it might suggest that you came and then went away again. The same would be true for 'came in'.

It would be more natural to say "I got here at 5am" instead of "I came here at 5am". To my ears it sounds unnatural to omit "here" in these two examples, you could say "came in" or "got in" though if you really didn't want to use "here". It is also fine to change the main verb to arrived, in this case "here" can be omitted: "I arrived at 5am" and "I arrived here at 5am" are both OK.