Do I use “past” or “passed” for this sentence?

past, and I do see the problem, because it is complicated and confusing which is which.  Past is a preposition when it has the meaning of "beyond" (as it does here).

 Past is the right word for that sentence.

Past. "How did you get past your anxiety."

Passed could be correct as the past tense of the verb to pass, but you can avoid this issue by using an alternative for passed such as over.

How did your anxiety pass?

How did you get past your anxiety?

I pass your house every morning on my way to work.I go past your house every morning on my way to work.When I passed your house this morning I waved at you. Once I was past the house I stopped waving.

past .............

'Passed' is a verb. Past tense of 'pass'. You can't "get passed". Unless you're a football or something.

You would use the word "past" in this sentence

"past" is correct in that sentence