If my research is addiction related, can I tell the professor I have first hand experience?

Yes you can. Even may.

The question is should you? What is the motivation? To brag your worldliness? To earn pity?

I advise no. Concentrate on learning the topic academically. Let your personal experience enrich your learning but more as deep background.

Your suitability as a candidate vis-a-vis the research area is very rarely dependent on a physical or psychological characteristic. For example, you don’t need to have Tourette’s syndrome to do good research into Tourette’s syndrome and some of the best researchers in prostate cancer don’t even have prostates. In fact, many countries have laws against using specific characteristics in selection processes.

I think that having personal insight into a particular object of research can certainly give someone empathy for subjects and this can come across in presentations or writings. However, that your research into addiction has been well-received is less likely to be due to your own experience as it is from the quality of your research — the rigour of your methods, the appropriateness of your analyses, the sensible ness of your inferences, and the weight of your contribution to the field.

I would strongly advise against your plan.