1. I went to hospital to have a health checkup.

2. I went to the hospital to have a health checkup.

3. I went to a hospital to have a health checkup.

(Which one is correct? Do I have to use article or not?)

If you're in the United Kingdom, then you'd probably use # 1. If you're in the U.S., both #s 2 and 3 are correct.