As long as I remember, one can travel in government buses too. these are long distance buses till north Andaman and baratang is a stop. but you might not be able to book them online, you can reach port Blair and book directly
To reach Baratang you can opt to take care there. This stretch can be done in about a little more than an hour by car. Port Blair to Baratang stretch can be done in about a little more than an hour by car.
You can hire local transport from Port Blair any transport agent. They will charge Rs.1700 per person for Baratang Trip including Breakfast and Lunch. This trip is include Jawora Ispand (Tribal Area, Baratang Islang, Lime Stone Cave, Mud Volcano. They will pick you up from your hotel and drop you to your hotel. Very adventurous this trip. They will pick up around 3.00 am at morning and drop you around 5.00 pm at evening. This is a full day trip at Baratang.