As a single word, checkup is a noun. As two words, check up is a verb. When hyphenated words, check-up is an adjective.

What is a sentence for check up?

She goes for a checkup every year. The doctor gave me a checkup. She brought in her car for a 30,000-mile checkup.

Is checkout or check out?

Check out is a verb phrase that means to sign for something or to observe something. Checkout means a place to buy things or the process of leaving a hotel as a noun, and it describes qualities of these things as an adjective.