anything

You should use "anything" in negative sentences:

I don't think we need anything.

... or in questions

Have you heard anything about the new Angelina Jolie movie coming out?

...or in clauses that start with "if":

If you see anything interesting, let me know.

In the last situation, you can also use "something":

If you see something interesting, let me know.

(something) comes up

When something "comes up", it means that something happens. But you use the phrase "comes up" to talk about things that happen which make you not show up or make you late to meet someone. For example:

I'm so sorry I'm late. Something came up with my kids' babysitter.

This means that something important happened that made you be late.

In America, especially among young people, some people don't like to make definite plans too far ahead of time. Instead, they like to make tentative plans and then discuss the details later. If you want to let someone know that you think the plan is definite, one way to do that is to say:

Let me know if something comes up between now and then.

This means "Contact me if you're going to change the plan for us to meet."

let (someone) know

"Let me know" means "tell me". But "let me know" is a more polite and friendly expression. It's very common to hear in an English-speaking workplace:

If anything comes up between now and then, just let me know.

You can also use it with friends:

Let me know when you're finished with the computer.