Don't forget to be on time because he's really strict (and he'll get angry at you)
Remember to be on time because he's a real shtickler (for timeliness) (and you'll get in trouble).
Make sure you're on time because he's a real hardass (and you'll get yelled at /written up).
Since you're saying to be on time, we already know that is what the person is strict about. So no need to say it twice. Though I do feel "shtickler" is usually used with "for N". It just means a person who is strict about something. Also, the "get angry/in trouble" part is implied and can be omitted (in fact, it's more natural and less redundant this way).