As a flight attendant I can tell you that getting your child his own seat so he can use his car seat is not only the absolute safest way to travel with him, but will also make the flight go much easier for you, your child, and those around you. I do realize however, that it may not be an option because of money. Having said this, I would recommend bringing a car seat with you anyway, just in case there are available seats left. If you talk to the gate agents and flight attendants they will usually try to accommodate your child in an empty seat, so they can use their car seat. If no seats are available they will gate check the car seat for you. As for using your baby bjorn, they will not let you use it for take off and landing because it is not an FAA approved restraint device. You should be able to use it at other times during the flight though. I also wanted to address the post that stated the flight attendants will give you a special seat belt to use with yours. Not all airlines have this (the one I work for does not) and it is my understanding that that is something usually found on international airlines, not domestic airlines, though I'm not sure about that. Just don't expect to get one.
Aside from this professional advice, I think the other posts are right that you just need to make sure you have plenty for him to eat, drink, play with, and distract him with. Letting him walk around is a great idea, just make sure the seat belt light is not on, as it is not safe for him (or anyone) to walk around during this time since you could encounter rough air that could make him fall and hurt himself. Plus, the flight attendants will probably make you sit down during this time anyway. Also, from my experience DVD players are fantastic at distracting little ones for long periods of time, as long as you are o.k. with using that.
Finally, don't worry too much about it. A 2-3 hour flight (as long as you don't have delays...pray for good weather) should be over fairly quickly. If people around you are not accommodating or are annoyed, just ignore them. They are probably just miserable people anyway. You will survive this! Also, let me know if you have any other questions. I'm happy to help any way I can.