Do brain tumor headaches normally just come and go?

They can. Headaches are a common complaint in people who have brain tumors. The headaches can even wake you up at night which is unusual for most headaches. Steroids can help reduce swelling around some tumors which can improve headaches in some people.
They vary. Headaches can be a presenting symptom of brain tumor although more than 98% of the headaches have nothing to do with brain tumor. The headaches from brain tumors are progressively more severe and often wake you up in early mornings. They are often followed or associated with nausea and vomiting and sometimes, seizures.