Learn Spanish (Latin America): Tengo un dolor de cabeza.

Both ways are perfectly natural! Tengo un dolor de cabeza translates literally to "I have a pain of head" (I have a headache). You could also use the verb doler from this lesson and say Me duele la cabeza (My head hurts me). They mean exactly the same thing in everyday conversation.

Is dolor a verb here?

No, dolor is a noun meaning "pain" or "ache". You can tell because it comes right after un (a/an). The related verb form that works like gustar is doler (to hurt).

Why is it dolor de cabeza and not cabeza dolor?

In Spanish, you can't stack nouns together to make a new word like we do in English with "headache". Instead, you connect the two nouns with de (of). So, a headache is always a dolor de cabeza (pain of head), and a stomachache is a dolor de estómago.

Do I always have to use un (a) before dolor de cabeza?

Not always. It is actually very common to drop it and just say Tengo dolor de cabeza (literally, "I have head pain"). Adding un just makes it "I have a headache." Both are completely correct and natural.