Jee main registration is live but I don't know if I have an option of choosing exam centre. Because I live in kota for prep but my hometown is in Bihar. And admit card is released just before the exams, it will be tough for me to catch.

Yes, during JEE Main registration, you can choose up to four preferred exam cities in order of priority. Since you are staying in Kota but belong to Bihar, you can select both locations. For example, Kota as your first choice and your hometown city in Bihar as another option.
When the admit card is released (usually a week before the exam), the final center will be allotted from your chosen options.